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Importance of the Mitzva of Milah
The mitzva of milah is equivalent to all other mitzvot in the Torah combined.
Without the mitzva of milah, the world would not exist.
In the merit of bris milah Hashem said that He split the sea for the Jewish people 
In the merit of brt milah, Aharon Hakohen entered the Kodesh Hakodashim every year.
The Night Before
In sephardic communities, it is customary on the night before a Berit Mila to assemble ten men in the home, as well as Torah scholars, to read special passages from the Zohar, in order to give the baby protection. There is value to reading Zohar even if one doesn't understand it.
During this event, the mother should come to the side room and recite Birkat Ha’gomel.
One does not need to eat bread at the meal for the night of the bris, and if one chooses to do so it would not qualify as a seudat mitzva, and thus would not allow for eating meat during the nine days. 
If the bris is pushed off past the 8th day, one should do the learning the night before the bris instead of on the 8th night.
If the bris is on Shabbat morning, one should do the learning on Thursday night, and if possible to gather ten people to learn on Friday night.
Obligation of the Mitzvah of Brit Milah
A father is obligated to perform a Brit Milah (circumcision) on his son or to appoint a pious Mohel to do so on his behalf. 
The father of the baby should stand near the Mohel (the one who is performing the circumcision) in order to show that the Mohel is his proxy. 
When a Brit Milah should take place
The Brit Milah is performed on the eighth day after the baby's birth.
The Brit Milah may be performed any time during the day, after HaNetz HaChama, however, one should make an effort to fulfill the mitzvah of Brit Milah early in the morning because of "Zarizin Makdimin LeMitzvot". 
If it will cause a fight among the family to perform the Brit Milah very early, one may delay it, but one should still try to make it as early as possible.
If there is a concern of a health risk to the baby, the Brit Milah should be delayed until the baby is healthy. Once the baby is healthy, in the case of some illnesses the Brit Milah should be performed without delay, while in other cases, the Brit Milah is only performed after 8 days after the baby became healthy.  Needless to say, because of the complexity of these issues, it is incumbent for a person to consult an Orthodox Rabbi to assess the situation. For more details about a delayed Milah, see the #A Delayed Milah section.
If a baby is born during Bein HaShemashot, since there is a doubt whether it was day or night, the Brit Milah is performed on the ninth day from the baby's birth. Because of the complexity involved in practically determining Bein HaShemashot an Orthodox Rabbi should be consulted.
Brachot for the Brit Milah
The father of the baby makes the Bracha "ברוך אתה ה' אלוקינו מלך העולם אשר קידשנו במצותיו וציונו להכניסו בבריתו של אברהם אבינו" (“Baruch Atta Hashem Elokenu Melech HaOlam Asher Kiddeshanu BeMitzvotav VeTzivanu LeHachniso BeBrito Shel Avraham Avinu”) before the Milah  This bracha should be said standing. 
The Mohel makes the Bracha "ברוך אתה ה' אלוקינו מלך העולם אשר קידשנו במצותיו וציונו על המילה" (“Baruch Atta Hashem Elokenu Melech HaOlam Asher Kiddeshanu BeMitzvotav VeTzivanu Al HaMilah”) prior to the Milah. The text is the same whether it is the father or the mohel reciting it. This bracha should be said standing. 
According to Sephardim a Shehechiyanu is recited at the Brit Milah. According to Ashkenazim a Shehecheyanu is not said at the Brit Milah except when a father is doing a Brit Milah for a firstborn son who is obligated in Pidyon HaBen. Today the Ashkenazic minhag is not to recite Shechiyanu outside of Israel.
Next Hagefen is made on wine and then the bracha of Asher Kidesh Yedid MeBeten.  The text is אשר קידש ידיד מבטן, אל חי חלקנו צורנו צוה להציל ידידות שארנו משחת. The one who recited this bracha should taste the wine before the paragraph of "אלוקינו ואלוקי אבותינו קיים את הילד וכו'" and naming the baby. It is sufficient to taste the wine and not drink a cheekful.
When the one reciting the bracha says ואומר לך בדמייך חיי the mohel takes a drop of wine and places it on the lips of the baby.
When reciting the bracha of אשר קידש ידיד מבטן Ashkenazim have the Sandak stand and hold the baby (Amidah Lbrachot), while Sephardim have the Sandak continue to sit.
After the bracha of אשר קידש ידיד מבטן there is a prayer said for the welfare of the baby and the mother and in that paragraph the baby is named.
If the Brit Milah takes place on Shabbat, if the one making the Brachot didn't yet make Kiddush, he should drink from the cup of wine a Melo Lugmav (cheekful) and another Revi'it. Sephardim hold that altogether he should drink a Reviyit or a Melo Lugmav.
The father of the baby and mohel don't need to close their eyes or cover the nakedness of the baby when reciting the brachot.
The father of the baby should place the baby on the lap of the Sandak. 
According to Ashkenazim, one shouldn't appoint the same person as a Sandak if he was already a Sandak for another one of one's sons.
Brit Milah Ceremony
When they bring in the baby they should say Baruch Haba Beshem Hashem out loud. The father of the baby says the pasuk אשרי תבחר ותקרב ישכון חצריך and those standing there answer נשבעה בטוב ביתך קדוש היכלך.
Everyone in attendance of the Brit Milah should stand except for the Sandak who is holding the baby.
After the makes the Bracha of LeHachniso, the people in attendance should answer Amen and add "כשם שנכנס לברית כן תכניסהו לתורה ולחופה למצוות ולמעשים טובים" - "Keshem SheNichnas LaBrit Ken Yikanes LeTorah ULeChpah ULeMaasim Tovim". Sephardim have the practice to say "כשם שהכנסתו לברית כן תכניסהו לתורה ולחופה למצוות ולמעשים טובים" unless the father isn't there in which case they say "כשם שנכנס לברית כן תכניסהו לתורה ולחופה למצוות ולמעשים טובים".
The mohel places the baby on the Kiseh of Eliyahu and says זה כסא של אליהו הנביא מלאך הברית.
Some have the practice to have candles lit in honor of the Brit Milah.
The minhag is to bury the orlah skin in dirt. Some say ונחש עפר לחמו when doing so.
Immediately after the brachot of the father the Sephardic minhag is to recite the 13 middot of Rachamim.
At the end of the milah the Sephardic minhag of Yerushalayim is to recite שיר המעלות אשרי כל ירא ה' and Kaddish Yemey Shelemah.
The Meal of the Brit Milah
It is a proper practice to make a meal on the day of a Brit Milah and it is considered a Seudat Mitzvah. 
The practice is not to invite people to this meal because there is an issue for someone who was invited to a meal of Brit Milah and didn't attend.
Ideally the meal should have meat but one fulfills the minhag even with chicken or fish.
According to Ashkenazim, one may perform a delayed Brit Milah on Thursday or Friday even if it will cause a situation of Pikuach Nefesh, however, Sephardim hold that one should not perform a delayed Brit Milah on Thursday or Friday. 
Everyone agrees that if a baby boy is born during the Bein HaShemashot (halachic twilight) on Wednesday can have the Brit Milah on Thursday next week. 
Milah On Shabbat
Sephardim hold that when there is a milah on Shabbat they shouldn’t have one mohel do the milah and another do the priyah. There is a dispute if it is true for Yom Tov as well.</ref>Otzer Brit v. 2 p. 379 writes Yam Shel Shlomo Yevamot 8:3 is machmir even on Yom Tov but we don't follow it (Petach Habayit). Or Yisrael p. 308 n. 42 writes it is asur even on yom tov. Yalkut Yosef Shabbat v. 4 p. 326 331:14 we hold that you shouldn't have two mohalim. It is clear from the footnote that it applies also to yom tov.</ref>
A father can do a brit on Shabbat even if he could have someone else do it.
Procedure of Milah
There are two parts to every Milah, the milah and the priyah. Milah entails cutting the Orlah skin which covers the Atarah and priyah entails peeling back the thin membrane beneath that skin until the Atarah is uncovered. 
The last stage of milah is metsisah. There is a great controversy whether this is considered a ritual part of the milah procedure or simply a medical practice of the days of the gemara and wouldn't be necessary today.  There is a further discussion if metsisah should be done with one's mouth or a utensil.  Because of the danger involved, many poskim hold that one should not and may not do metsisah with one's mouth.
The Gemara Kiddushin 29a states that a father is obligated to perform a Brit Milah for his son. Some rishonim assume that the entire obligation on the father is to make sure that the Milah takes place but he doesn't need to personally perform it nor even appoint a Mohel, as long as it is taken care of. These rishonim include the Maharach Or Zaruah (responsa n. 11) and Tosfot Rid (Kiddushin 29a s.v. Otto). However, most rishonim assume that there is an obligation upon the father to actually perform the milah and if he doesn't know how, he should appoint a Mohel to do it for him. These rishonim include the Rambam (Milah 1:1), Darkei Moshe (YD 264:1), and Rosh Chullin 6:8.
The Ohr Zaruah (v. 2, Siman 107) writes that if the father knows how to do the Milah and nonetheless asks a Mohel to do it for him, he is losing out on his mitzvah. Shach CM 382:4 agrees. The Darkei Moshe 264:1 argues that we always use the principle of agency (Shaliach Adam KeMoto, Kiddushin 41a) and this mitzvah should be no different. The Kesot CM 382:2 defends the Ohr Zaruah saying that Brit Milah is a mitzvah that needs to be done with your body, and can't be delegated to be done by someone else.
↑Pesachim 4a says that the Brit Milah may be performed any time during the day but those who have alacrity in fulfilling mitzvot will do it early. The gemara learns this concept of Zarizin Makdimin LeMitzvot from Avraham Avinu when he went to perform the Akeda early in the morning. Shulchan Aruch YD 262:1 codifies this as halacha.
↑ Shulchan Aruch YD 265:1 writes that this Bracha should be made after the cutting of the Orlah before the Periyah, however, Yalkut Yosef (Sova Semachot vol 2 pg 60) writes that the minhag is to make it before the Milah and the Birchat Hamilah altogether. Yalkut Yosef Milah p. 77 writes that the minhag of Israel, Bavel, and Eastern Sephardi countries is to recite Lhachniso before the Milah. Even though Shulchan Aruch holds like the Rosh (Shabbat 14:10, Teshuva 26:1) that it can be said afterwards, the Rambam (Rav Avraham Ben HaRambam cited by Yabia Omer OC 2:17:7) and Rif (Teshuva 293) hold it should be said beforehand.
↑Rav Mordechai Eliyahu's comment on Shulchan Aruch Kitzur 163:8, Bayit HaYehudah vol. 10 pg. 182, see also Rabbi Aryeh Lebowitz
↑Yalkut Yosef Sova Semachot 2:11:3 based on Taamei Haminhagim writes that the meal should have meat initially but if that's not possible it could have chicken or fish. Rabbenu Bechay Beresheet 21:8 writes that the minhag to have a seuda after a Brit Milah is based on the party Avraham made for Yitzchak and the midrash writes that the party was for fact he was able to give Yitzchak a Milah.
↑Otzar Habrit 3:17:4 p. 239 writes that even if there wasn’t a minyan present at the seudat brit milah the seuda is still a seuda and the minyan isn’t the halacha but a minhag. Bechorat Habrit 265:66 says ten is enough but it isn't critical. See also Maharam Shik YD 386.
The Gemara Shabbat 19a quotes the Briatta which states that one may not board a boat in order to travel on Shabbat unless one got on before 3 days before Shabbat (whether or not this includes Wednesday, see Mishna Brurah 248:4).
The Rif (Shabbat 7a-b) explains that the reason for this restriction is that if one boards a boat within 3 days of Shabbat it will negatively impact the passenger’s oneg Shabbat (enjoyment of Shabbat), however, if it’s started earlier the passengers will get used to it and be able to enjoy Shabbat. The Rosh (Shabbat 1:38) and Rambam (Shabbat 30:13) agree with this reason.
However, the Baal HaMoer (on Rif) explains that the reason that one may not board a boat close to Shabbat is because it’s likely that there will be life threatening danger which will necessitate a violation of Shabbat. However, when it is begun earlier there’s no issue because the obligation to prepare not to violate Shabbat even for life threatening danger doesn’t begin (explained by Mishna Brurah 248:8).
The Rashbatz 1:21 originates based on the Baal HaMoer’s concept that a convert shouldn’t schedule his Brit Milah on Thursday because it will lead to a violation of Shabbat (if he doesn’t recover fully) and the same would be true of a baby who had his Brit Milah delayed. This is codified by the Tur and S”A YD 268. [Interestingly, the Taz explains that even if there’s no concern of a violation of Shabbat there’s an issue of ruining oneg Shabbat and so it would be forbidden to do the delayed Brit Milah on Thursday or Friday.]
However, the Shach YD 266:18 argues on the Rashbetz that the Baal HaMoer would permit just like it’s permitted to board a boat on Friday for the purpose of a mitzvah (S”A OC 248:1) and Brit Milah is a tremendous mitzvah. The Magen Avraham 331:9, Mishna Brurah 331:33, and Sh”t HaRanach 38 (quoted by Tzitz Eliezer 12:43) agree with the Shach.
The Chida in Birkei Yosef 248 quotes several achronim who defended the Rashbetz by saying that it was only permitted to board a boat on Friday for a mitzvah if one stipulated with the group that they wouldn’t continue to travel on Shabbat (S”A 248:1) and since there’s no way to make such a stipulation regarding Milah it’s forbidden to do it within 3 days of Shabbat like the Baal HaMoer.
Sh”t Yabia Omer YD 5:23, Menuchat Ahava (vol 1, 1:6), and Rav Mordechai Eliyahu (comments to Kitzur Shulchan Aruch 163:4) accept the Chida as halacha.
However, the Sh”t Tzitz Eliezer 12:43 rejects the Chida because he points out that the Tashbetz himself wasn’t sure whether to hold like Rebbe that one must stipulate or Rabben Shimon and one wouldn’t have to stipulate and only as a stringency did the Tashbetz hold like Rebbe. The Tzitz Eliezer concludes that it’s illogical that the Tashbetz would have postponed the Milah from Thursday just because of a stringency of holding like Rebbe. [See further in the Birkei Yosef who gives a second defense of the Rashbetz and Tzitz Eliezer who rejects it as well.]
↑Menuchat Ahava (vol 1, 1:7) and Sh”t Yabia Omer YD 5:23(4) explain that since the Brit Milah’s original time was set for Thursday because of a doubt it’s not considered like it was delayed specifically for that time.
↑The Mishna (Shabbat 133a) outlines the 3 stages of milah including milah, priyah, and metsisah. For a discussion of metsisah see further. Rashi s.v. mohalin refers to cutting the orlah skin. Rambam (Milah 2:2) elaborates that the orlah skin is the skin that covers the Atarah. Rashi s.v. veporin writes that priyah is peeling back the membrane covering the tip of the gid. Rambam (Milah 2:2) elaborates the priyah is the membrane underneath the orlah skin.
↑ The Mishna (Shabbat 133a) writes that metsisah is the last stage of milah. The gemara (133b) comments that the reason that is practiced on Shabbat is because if it isn't done there is a critical danger to the baby's life. The Rambam (Milah 2:2) echoes this and says that metsisah is for medical reasons. The Tiferet Yisrael (Boaz Shabbat 19:1) writes that the entire purpose of metsisah is medical and if nowadays it causes a medical danger it doesn't need to be done. See also the Maharam Shik (OC 152) takes the position that there is no danger with metsisah. However, the Mishkenot Yacov (YD 63) writes that drawing blood (hatafat dam) is an integral part of the ritual milah. He proves this from the Zohar and Rashi (Shabbat 134a s.v. lekarchei).
↑Har Tzvi (YD 214) writes that since there are dangers with doing metsisah with one's mouth, one may do it with a utensil.
↑Rav Hershel Schachter in a shiur on yutorah.org ("Hilchos Shabbos 3" min 27-30) said that there is a real danger to the life of the baby with doing metsisah with one's mouth and it is foolish to continue doing it today. A number of poskim are quoted in an article by David Shabtai and Raymond Sultan entitled Medical Risk Taking in Halacha in 'Journal of Halacha and Contemporary Society, 2006. See also the Binyan Tzion 23 who writes that a mohel who does metsisah must be certain that he is not infected with herpes and cause the baby to become infected. Yalkut Yosef Sova Semachot v. 2 ch. 6 fnt. 1 writes that one may use metzitza bkli such as with a tube if there's a concern of danger, otherwise it is done with one's mouth as the Sdei Chemed (Kuntres Metzitza) writes.
With respect to the general question of relying on doctors to ascertain that there's no danger see Rav Kook's teshuva about metzitza in Daat Kohen YD 140 who argues that we can never rely on doctors to make such a claim. Yabia Omer OC 7:53 likes this idea.