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The proper way to escort out the Shabbat is to have a festive meal after it is over, just like it is welcomed in with the utmost respect. Just like one would escort a King out of his city with honor, so too Shabbat.
- A person should set his table on Saturday night even if one only plans on eating a Kezayit in order to honor the completion of Shabbat. Even though many aren’t careful about it, a Yireh Shamayim (a G-d fearing person) should make an extra effort to keep it.
- One should eat with specific intent that it should be for the meal of Melaveh Malka, preferably saying this expressly. See further in the topic on Having Kavana for Mitzvot.
- At the end of Birkat HaMazon for Melava Malka, some authorities hold that one should say say "migdol yishuot malko" like on Shabbat itself, instead of "magdil yishuot," while others say that one should say the regular "magdil yishuot" on Melava Malka just like during the week.
- Some poskim allow eating meat leftovers on motzaei Shabbat even during the nine days while some poskim say that it's prohibited. See Nine_Days#Shabbat_and_Leftovers for the sources.
Who is Obligated?
- Many say that women are also obligated to eat Melava Malka.
- Kaf HaChaim quotes poskim who write that women should be having the Melava Malka seudah in order to sustain the Luz bone. He also quotes in the name of Reb Elimelech that the seudah of motzei Shabbos is a segulah (good sign) for women to have easy childbirth.
When Should One Perform Melava Malka?
- Melava Malka should be fulfilled as soon as possible after Shabbat. If one isn’t hungry, one may have it later but shouldn’t do an established activity before having Melava Malka. However, certainly one shouldn’t delay it past Chatzot of the night. However, if one did delay past Chatzot one may still fulfill it until Alot HaShachar.
- If one extended his Seudat Shelishit until it got dark, some say that one is exempt from having Melava Malka, and some argue. 
- There’s no obligation for having Melava Malka on Motzei Yom Tov.
What Should One Eat for Melava Malka?
- One should try to eat bread for Melava Malka, but if he cannot then cake is sufficient. If he cannot even eat cake, he should at least have some fruits.
- One should wear nice Shabbat clothes during this meal.
- Some have the custom to light candles and sing special zemirot on motzei Shabbat after Havdalah. 
- One should spread a tablecloth on the table and set the table nicely.
- It is beneficial to consume hot food and drink, and wash with hot water, as these provide refuot (health benefits). 
- Rambam Hilchos Shabbos, 30:5
- Shulchan Aruch O.C. 300:1 writes that a person should set his table for Malveh Malka on Saturday night. This is based on the Gemara Shabbat 119b. Tur 300:1 says that this is the proper way to escort the Shabbat out just like we bring it in with proper respect. Mishna Brurah 300:1 explains that although it is important, Melava Malka is not quite as important as the three Shabbat meals, meaning if you can't afford to eat Melava Malka unless you eliminate one of the Shabbat meals you shouldn't do that. Mishna Brurah 300:2 explains that the reason is to give respect to the exiting of Shabbat.
- Aruch HaShulchan 300:3, Shemirat Shabbat KeHilchata 63:1. The Midrash Kohelet Rabba 12:5 says that in the future, a person will be resurrected using a bone located in one’s spine. The Kaf Hachayim 300:1-2 says that this bone called the luz is fed only with food eaten at a melave malka. Mishna Brurah 300:2 quotes the Taz saying the same thing but calls the bone the nisko. In Shaar Hatziyun 300:7, he explains that since this bone didn't get pleasure from Adam's sin, this bone was spared the curse of death.
- Mishna Brurah 300:1
- Kaf Hachaim 189:11
- Yalkut Yosef (Kitzur Shulchan Aruch 300:4)
- Machatzis Hashekel 300:2, Yechave Daat 4:25, Shemirat Shabbat KeHilchata 63:3. Magen Avraham 291:15 quotes the Avudarham saying that some women had the custom not to do chores on motzaei shabbat. The Gra in Maaseh Rav 150 says that he was also very strict on his wife. The Pri Megadim A”A 300:1 says that maybe they’re not chayavos based on their safek by havdalah.
- Kaf HaChaim 300:2
- Kaf HaChaim 300:4
- Mishna Brurah 300:2 and Shemirat Shabbat KeHilchata 63:5 write that Melveh Malka should be done right after Shabbat and shouldn’t do a established activity before Melvah Malka. However, they add, at least one should eat Melava Malka by Chatzot because the Shaare Teshuva 300:1 quotes the Arizal that the holiness of shabbat only remains until Chatzot. Nonetheless, Yalkut Yosef (Shabbat vol 1 pg 630) adds that if one didn’t do it before Chatzot one may still fulfill this mitzvah until Alot HaShachar. One of his proofs is from a story about the Gra in Tosefet Maaseh Rav (39). One time the Gra was sick on Saturday night and didn't eat Melava Malka, when he got somewhat better it was still before Alot Hashachar and so he ate Melava Malka. Kaf Hachayim 300:14 says it should lechatchila be eaten within four halachic hours of shabbat.
- Eliyah Rabba 300:1 writes that he heard that if Seudat Shelishit until it got dark one is exempt from having Melava Malka. However, Tehillat David 300, Kaf HaChaim 300:11 disagree with this.
- Shemirat Shabbat KeHilchata 63:4 rules that there's no obligation to eat Melveh Malka on Yom Tov. See also Sh”t Rivivot Efraim 3:292 who writes that the Chazon Ish would have a Melava Malka on Motzei Yom Tov.
- Magen Avraham 300:1, Mishna Brurah 300:1. Yalkut Yosef (Shabbat vol 1, 300:1). The Gra in Biur Hagra 300:1 says that bread must be eaten at the seuda. In Maaseh Rav 150 it says that the Gra was extremely strict about this halacha. It tells a story about when Rav Chaim Volozhin came to visit him and hadn't ate melava malka because he wasn't feeling well and the Gra told him to eat. Rav Moshe Shternbuch in Teshuvot vihanhagot 2:166 explains that the reason the gra was so strict on this is because he felt it was something that was very neglected.
- Kaf HaChaim 300:6, Shemirat Shabbat KeHilchata 63:2. Leket Yosher page 85 says that one shouldn't remove his shabbat clothes until he goes to sleep on motzaei shabbat.
- Mishna Brurah 300:3
- Mishna Brurah 300:1. Rav Moshe Shternbuch in Teshuvot vihanhagot 2:166 says that setting the table shows that we really want to extend shabbat and aren't happy that it is over.
- Gemara Shabbat 119b See Rashi there for חמין and מלוגמא.