There is a mitzvah to wash one's hand after a bread meal before Birkat HaMazon. This mitzvah is called Mayim Achronim. The rabbis loosely derived it to a pasuk “VeHayiteem Kedoshim” - we should be holy. Additionally, Mayim Achronim was instituted in order to prevent Sedomite salt from getting into one’s eyes, which could potentially blind oneself. 
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The basics of the obligation
- The Sephardic custom until this day is to wash one's hands after a bread meal. This practice is called Mayim Achronim. It is based on many Rishonim who hold Mayim Achronim is an obligation nowadays either because there is a concern of Sedomite salt even in our times, there is an obligation of being Kadosh through observing Mayim Achronim, or there is an obligation of cleaning one's hands for Birkat HaMazon. 
- The Ashkenazic minhag is not to wash Mayim Achronim, however, some say that it is preferable to wash Mayim Achronim.
Who is obligated in Mayim Achronim?
- Both men and women are obligated in Mayim Achronim.  The custom of many Ashkenazic women, however, is not to wash mayim acharonim. 
- Even if one didn’t touch the food with his hands, one is obligated in Mayim Achronim. 
- Some say that one is obligated to wash Mayim Achronim even if one didn’t even eat a meal but just touched salt. However, the minhag is not to wash after just touching salt. 
- Even Ashkenazim who are not stringent concerning Mayim Achronim, if one’s hands are dirty and one would be concerned with their cleanliness, one is obligated to clean one's hands or wash them before Birkat HaMazon. 
- It is proper to train children to wash mayim acharonim as well. 
Interruptions between Washing and Making Birkat HaMazon
- Some are strict not to talk even words of Torah. If one spoke it’s preferable that one wash again.
- Some have the custom to say Shir HaMaalot, Lamnaseach, or Avarecha after Mayim Achronim since it’s a necessary element to Birkat HaMazon.
- One can answer amen, kaddish, kedusha and barchu.
- Preferably, one shouldn’t wait or do actions unrelated to Birkat HaMazon more than 12 seconds; those who wait for everyone to wash their hands have what to rely on. 
- In our time that everyone reads Birkat HaMazon one should try to have multiple vessels so everyone doesn’t have to wait long until Birkat HaMazon.
- If there's only one mayim achronim vessel to wash with, if there's five people or less, the one who is going to lead the zimmun should wash first so that he has time to prepare the text of the four brachot of birkat hamazon. If there are more than five people, then the least dignified should wash first and they should continue to wash in the order of how they're sitting and the fifth to last person should be the would who is going to lead the zimmun.
Water that is fit for Mayim Achronim
- One shouldn’t use scalding hot water, rather lukewarm water is permitted and it preferable to use cold water. Hot water that cooled is also permitted. 
- Bitter, dirty, or smelly water not fit for animals to drink is permitted. 
- Saliva is unfit; if nothing is available is better than nothing. 
- If there’s no water, one can use other liquids but not wine, yet if nothing else is available then one should wash with wine. 
Requirement of actually pouring the water
Amount needed for Mayim Achronim
How the washing is done
- One should wash up to the second set of knuckles on one's fingers and on one's thumb one should wash up to the first knuckle. It’s preferable to be strict for the opinion that one should to wash the entirety of one's fingers. 
- There’s no Bracha for Mayim Achronim. 
- One should face his fingers downward so that the dirt falls off. 
- Only one washing is required, not two or three times. 
- Although it is not required to use a vessel to pour the mayim acharonim, it is preferred to do so.
- One may not wash over the ground because “ruach rah” hovers over it, rather one should wash into, in order of preference, a vessel, twigs or stones on the ground that absorb water, tiled flooring, ground not used like under the table and certainly a sink which has a drain. 
Drying one's hands
- From the strict law, one doesn’t have to dry his hands. Nonetheless, it’s preferable to be strict to dry one’s hands. 
- Some have the practice to say “Mayim Achronim Chovah (is obligatory)”. 
If one forgot Mayim Achronim
- Brachot 53b, Chulin 105a-b
- Rambam (Brachot 6:3), Rabbeinu Yonah (Brachot 40b), and Rif (Chulin 37b) hold that it applies because of danger that the Sedomite salt can cause. Raavad (Brachot 6:2), Rashba (Chulin 105a), and Rashbetz (Brachot ch. 8) seems to hold that the purpose is to have clean hands for Birkat HaMazon. Shulchan Aruch 181:1 rules that it is an obligation to do Mayim Achronim even nowadays.
- Tosfot (Brachot 53b, Chulin 105a, Eiruvin 17b) hold that there's no obligation to wash Mayim Achronim nowadays because the concern of Sedomite salt is not longer relevant. Rama in Darkei Moshe 181:1 and Levush 181:9 codify the opinion of Tosfot and add that that is the Ashkenazic Minhag. The Gra 181:12 and Aruch HaShulchan 181:5, however, hold that it’s preferable for Ashkenazim to wash Mayim Achronim. Mishna Brurah 181:22 first explains the opinion of Tosfot and then cites a number of achronim who are strict about washing Mayim Achronim.
- Torah Chaim Sofer 181:1, Sh"t Shalmat Chaim 119, Sh"t Yitzchak Yeranen 4:35, Sh"t Rivovot Efraim 1:140(3), Sh"t Sh"t Divrei Chachamim (pg 62) in name of Rav Elyashiv, Rav Ovadyah Yosef in Halichot Olam (vol 2 Shalach 1), Halacha Brurah 181:1, Hilchot Bat Yisrael (pg 59) in the name of Rav Shlomo Zalman Aurerbach hold that women are obligated in Mayim Achronim.
- Rabbi Yitzchak Yaacov Fuchs in Halichos Bas Yisroel (pg 59): HaGaon Rav Shmuel HaLevi Wossner writes in Shevet HaLevi, Vol. 4, Orach Chayim, No. 23, that the basis for women not washing mayim acharonim is found in the Shulchan Arukh, Orach Chayim 181:10. Chazal explain that in Talmudic days a certain type of salt, called melach s'domith (salt from Sodom) was commonly used and that it could cause blindness if it came in contact with the eyes. The original purpose of mayim acharonim was to rinse the melach s'domith off the hands. The Shulchan Arukh states that since this salt is no longer in use, the obligation of mayim acharonim no longer applies. Although many men follow the view of other authorities who maintain that the mitzvah does apply today, women commonly follow the view of the Shulchan Arukh. HaGaon Rav Yonah Merzbach reports that in many German communities, before the Second World War, women omitted mayim acharonim and that this practice was common even in households extremely scrupulous in the performance of mitzvoth.
- Halacha Brurah 181. If one has no water (up to 4 mil, which is less than 4 kilometers, on the way or 1 mil in the opposite direction) he should not touch the food when he eats.
- The Gemara Chulin 105b states that there's an obligation to wash one's hand after touching salt. The Rif (Brachot 8) codifies this gemara. The Talmidei Rabbenu Yonah, Raah, Michtam, and Rashbez also quote this gemara. The Rashba (Torat HaBayit 6:1) is lenient. However, the Tur and Shulchan Aruch didn't quote this halacha. Therefore, Pekudei Elazar 181 and Halichot olam vol 2 shalach 1 write that nowadays no one is strict on washing after touching salt.
- Shulchan Aruch 181:10, from Tosfot (Brachot 53b), Nemukei Yosef, Rosh 8:6, Mordechei Br 191, Aguda Br 8 and Tur. So too, if his ate or drank not in a bread meal and his hands are dirty he still must wash hands before the Bracha Achrona. Even someone who isn't finicky who touched an unclean area or a covered area on the body must clean or wash his hands in any way before making a Bracha. Magen Avraham 181:10, Zechur leAvraham 40, Mishna Brurah 181:23, Kaf Chaim 181:9, Pri Megadim A”A 9.
- Rav Chaim Kanievsky cited in Otzar Hamayim chapter 4
- Kesef Mishna (Brachot 6:20) infers from Rashi and Rambam that speech isn’t an interruption between mayim achronim and birkat hamazon. The Magen Avraham 165:4 questions it but seems to agree. On the other hand, the Arizal (cited by Bear Heitiv 179:1) holds that it is forbidden to talk betwen mayim achronim and birkat hamazon. This is also the opinion of the Drisha 179:2, Eliyah Raba 179:1, Pri Megadim E”A 179:1, Chayei Adam, Zachur LeAvraham, Magen Giborim, Maharsham in Daat Torah, and Mishna Brurah 179:1. The Shulchan Aruch HaRav permits 2 or 3 words and the Aruch HaShulchan is lenient like the Kesef Mishna. Bear Heitiv 179:1 cites the Magen Avraham as though he is lenient and Shaar Hatziyun notes this difficulty.
- Kaf HaChaim 157:22 in name of the Arizal and Or Tzaddikim, Petach HaDvir 181:2, Chida (Simchat Haregel Seder Birkat HaMazon), Tziporen Shamir 5:61, Chesed LeAlafim 181:7, Ben Ish Chai Shalach 15, against the Aruch HaShulchan 181:9.
- Tehilat LeDavid 181:2 says not to answer just like one juxtaposes Geula to Tefilah. However the Halacha Brurah 181:5 disagrees based on a Sfek Sfaka that maybe one can go like those who allow speech between mayim achronim and Birkat HaMazon and maybe one can go like those who say that one can answer devarim shbekedusha between Geula and Tefilah. Additionally it’s a mitzvah that’s fleeting.
- Bach 179, Pri Megadim E"A 179:1, Mishna Brurah 179:1 in name of the derech hachaim that one should rewash, Taz 181:3 says one can wait for 4 people to wash hands after you, so says the Lavush 181:6, Olat Tamid 181:4, Magen Avraham 181:5, Shulchan Aruch Harav 181:5 Mishna brurah 181:15, Kaf Hachaim 181:18, Peulat Tzaddik 3:71, Yabea Omer Y”D 2:1(11). The stringency of unrelated actions is sourced in the Ritva Hilchot Brachot 3:17 and 6:21 and a similar approach is cited by Kesef Mishna Brachot 6:20.
- Beiur Halacha s.v. Min Hakatan, Halacha Brurah 181:15. However, in earlier times, only one would say Birkat HaMazon. Accordingly, Shulchan Aruch O.C. 181:6 rules the one doing the Birkat HaMazon should be the first to wash but if there’s more than 5 people washing, then they begin washing from the least important person until there’s 5 people left and then the one doing Birkat HaMazon washes.
- The Gemara Brachot 46b establishes that the least dignified person should wash mayim achronim first because it isn't an honor to wait without talking between mayim achronim and birkat hamazon. However, the one who is going to lead zimmun should make sure to wash first if there's a small group of five or less, or fifth to last if there's a larger crowd so that he has time to prepare the text of the four brachot of birkat hamazon. Raavad Brachot 7:12 rules accordingly. However, the Rambam 7:12 explains that the greatest person always washes first. The Kesef Mishna explains that the Rambam thought that the gemara was only applicable if they would remove the personal tables after they washed in which case it isn't an honor to wash first, but now that the table isn't removed at all, the honor is to wash first so that you can have clean hands. Shulchan Aruch 181:6 rules like Shulchan Aruch. Halacha Brurah 181:15 agrees.
- The Rashba Brachot 46b s.v. bizman explains that it isn't considered an interruption for the one who is doing zimmun to wash fifth to the end because it is only a period of time between washing and birkat hamazon and also the interruption has a purpose. Mishna Brurah 181:15 agrees.
- Tur and S”A 181:3 from Chulin 105a, rule leniently by lukewarm water, Kesef Mishna says Rambam (Brachot 6:16) allows it even initially. Magen Avraham 181:3 says in name of Maharshal that specifically cold water should be used. Mishna Brurah 181:7 says if one only has lukewarm, one shouldn’t be stringent.
- Rashba Torat Bayit 6:2 says all water unfit for Mayim Rishonim is fit for Mayim Achronim. So is brought in the Bet Yosef (comments that so is implied by the Gemara and Posikim), Darkei Moshe, Orchot Chaim (Netilat Yadim 31), Kolbo (23) in name of Raavad, but the Riaz Netilat Yadim 8:20 is stringent. Many Achronim follow Rashba and Raavad such as Magen Avraham 181:7, Chaye Adam 46:2, Aruch Shulchan 181:6-7, Shulchan Aruch HaRav 181:8, Mishna Brurah 181:21, Kaf HaChaim 181:28.
- Peni Mabin O”C 38, Minchat Elazar 3:54, and Sherit Yehuda Blum 17 are stringent against the Torat Yekutiel Rosenberg O”C 6 who is lenient.
- Bet Yosef brings the Rabbenu BeChaye in Shulchan Shel Arba that allows all liquids and the Tur who doesn’t allow wine, S”A 181:9 holds leniently by wine, but many Achronim, such as the Levush 181:7, Atret Zekenim 181:2(2), Shulchan Aruch HaRav 181:8, Mishna Brurah 181:21 are strict like the Tur.
- Magen Avraham 181:1, Shulchan Aruch HaRav 181:5, Mishna Brurah 181:3, Aruch Shulchan 181:7 agaisnt the Eliyah Rabba 181:9 who is lenient. Kaf HaChaim 181:9 says it’s effective for cleaning but not for the kabbalistic reasons and Zivchai Zedek 3:35 holds it’s not forbidden.
- Raavad in Tamim Daim 66, Orchot Chiam Netilat Yadim 31, Kolbo 23, against Riaz Netilat Yadim 8:20
- The Rashba (Torat HaBayit 6:2) writes that there's no specific ammoutn one needs to use for Mayim Achronim. The Orchot Chaim Netilat Yadim 31 and Kolbo 23 in name of Raavad agree. This is codified by the Levush 181:7, Atret Zekinim 181:1, Shulchan Aruch Harav 181:4, and Kaf Hachaim 181:4. The Mishna Brurah 181:19, however, notes that the practice of Gra was to use a Reviyit. On the other hand, the Chida in Simchat Regel (Seder Birkat Hamazon) and Ben Ish Chai (Shalach 8) argue that one should try to use as little water as possible. Accordingly, says the Birur Halacha 181:13 writes that if one’s hands are dirty one should first wash them first and then wash them again with the intent that it satisfy the obligation of Mayim Achronim.
- Birur Halacha 181:4 from Bet Yosef 181:1, Eliyah Raba 181:1, Mateh Yosef O”C 2:18(32) in name of Orchot Chaim, and Yad Aharon Siman 158
- Based on Rashba Torat Bayit 6:1 and Raavad, the S”A 181:4 rules that one should wash one's fingers up to the second row of knuckles. Rabbenu Bechaye in Shulchan Shel Arba, however, requires one to wash one's whole finger. Mishna Brurah 181:10 quotes Talmedei Rabbenu Yonah who say that the thumb must only be washed up to the first knuckle. Halacha Brurah 181:11 agrees. Beiur Halacha s.v. Ad Perek Sheni and Kaf HaChaim 181:17 agree that it's preferable to be strict for the opinion of the Arizal that one should wash one's whole finger. Rabbi Mansour on Dailyhalacha.com quotes the Shulchan Aruch and says to preferably be machmir for the opinion of the Arizal (to wash the entirety of the fingers). Halacha Brurah 181:11 agrees.
- S”A 181:7. The Bet Yosef quotes the Ramah who hold that there is a Bracha and so holds some geonim and rishonim such as Yeraim 23, Shibolei Leket 149, Mordechai Brachot 191, Ravyah 1:145, Raavad Brachot 6:2. However the Bahag Brachot 6, Talmidei Rabbanu Yonah 40b, Rosh 8:6, Rambam Brachot 6:2 say not to make a Bracha.
- S”A 181:5 based on Sotah 4b. This is also codified by the Rambam (Brachot 6:16) and Tur.
- Bet Yosef in name of Mordechai (Brachot 207), Mishna Brurah 181:21.
- Sh"t Or Letzion 2:13:1, Levush 181:2, Eliya Rabba 181:3
- S”A 181:2 from chulin 105a and many rishonim that hold like it. The Magen Avraham 181:2 in name of Chinuch that anything stopping the water from pouring on the ground, but the Rambam (brachot 6:16) seems to only allow a vessel. The Gra 181:4 explains S”A that initially we follow Rambam but if there’s no vessel available then we can rely on the Rashba (Torat Bayit 6:2) and Tur (181:2). Yabia Omer O”C 5:2 permits a sink by washing in the morning and all the more so for Mayim Achronim. Birur Halacha 181:8 concludes that one can pour it on the ground after one washed into the vessel.
- S”A 181:8, Bet Yosef quotes Rashba in Torat Bayit 6:2 that one doesn’t need to dry one’s hands and the Rambam Brachot 6:20 who holds one needs to dry one’s hands. S”A brings both opinions but the Mishna Brurah 181:21 points out that in Siman 173 S”A implies that he holds Mayim Achronim doesn’t need drying.
- Rav Chaim Palachi in Ruach Chaim 181:1, Ben Ish Chaim Shalch 7 says that it can count as a word of Torah during a meal.
- Chesed Alafim 181:8, Ben Ish Chai Shalch 11, Kaf Chaim 181:26, Mishna Brurah 181:2 against the Ometz Yozfa 155 who says not to wash because it degrades the sanctity of Birkat Hamazon.