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A Mitzvah to eat:
Rich meat and sweet beverages: Throughout the meals of Rosh Hashanah the custom is to eat rich meat and drink honey [beverages] and other sweets as an omen that the year should be sweet and rich. [One should however avoid eating the meat of a goat as goats are rooted in Gevura.]
Not to fully satiate oneself: Nevertheless one should not eat to his full capacity in order so he not be led towards frivolous behavior, and so he keep the fear of G-d consciously on his mind.
Is there a Mitzvah of Simcha on Rosh Hashanah?
No. Hence one is not obligated to drink wine, or buy his wife clothing or jewelry on behalf of Rosh Hashanah.
Is one obligated to drink wine on Rosh Hashanah other than for Kiddush?
Although there is no obligation to drink wine due to the Mitzvah of Simcha on Rosh Hashanah, unlike other Holidays, nevertheless due to the Mitzvah of Oneg one is obligated to establish his meal on wine. This means that he is to drink wine in middle of the meal, as is the law by all Holidays. However there are opinions which rule that one is not to drink wine on Rosh Hashanah [other than for Kiddush]. Many Poskim argue on this opinion. Regarding the opinion of the Shulchan Aruch Harav in this matter see footnote.
May one fast until midday on Rosh Hashanah, or must he eat or drink something beforehand?
One is allowed to fast until midday on Rosh Hashanah. This normally occurs if one did not eat or drink anything in the morning and the prayers continued until past midday.
Diminishing in certain foods on Rosh Hashanah:
The Rashal writes that he received from his grandfather that he refrained from eating fish on R”H in order to diminish his physical pleasure in a specific matter and remember the day of judgment. This would prevent him from coming to light headed behavior.
 597/1; Michaber 597/1
 Admur ibid; Michaber ibid; Taz 597/1; Levush 597; Bach 597; Biur HaGr”a; Chayeh Adam 139/11; Shaagas Aryeh ibid; Kaf Hachaim 597/3 Other opinions: There are opinions which rule it is a Mitzvah to fast on Rosh Hashanah and Shabbos Teshuvah just like on any other day of Aseres Yimei Teshuvah. However the final ruling is not like this opinion and it is thus forbidden to fast on Rosh Hashanah and Shabbos Teshuvah just like on any other Shabbos and Yom Tov. [Admur ibid] This ruling [that it is a Mitzvah to fast] is recorded in the name of some of the Geonim and is brought in Terumos Hadeshen 278; Tur 597; Beis Yosef 597. The Taz 597/1 however proves that there is no Mitzvah to fast on Rosh Hashanah as the verse describes how Elisha ate on Rosh Hashanah by the house of the Shunamis. However see Elya Raba 597/1 and Birkeiy Yosef 597/1 which ask on the proof of the Taz ibid. The Shita Mekubetzes [Beitza 2, brought in Kaf Hachaim 597/4] writes in the name of the Ritva “One is allowed to fast [not an obligation] on R”H or Shabbos Teshuvah for the sake of Teshuvah being that it is a day of judgment. On him it says that one who fasts on Shabbos Hashem tears a decree of 70 years. However this may only be done by men of stature and not by the masses as this can lead them to belittling the Yom Tov of R”H”. The Machazik Bracha 597/1 concludes that practically no person may rely on this and rather he is to fast on the weekdays of Aseres Yimei Teshuvah. Likewise the Shaagas Aryeh [101-102, brought in Shaareiy Teshuvah 597/1 and Birkeiy Yosef ibid] proves that one must eat on Rosh Hashanah just like any other Holiday and it follows the same laws of Kiddush and Hamotzi as any Holiday. Practically majority of Poskim rule that one may not fast on R”H and hence the ruling follows the majority opinion. [Kaf Hachaim 597/3; So rules: Michaber ibid; Admur ibid; Taz ibid; Levush 597; Bach 597; Biur HaGr”a; Chayeh Adam 139/11; Shaagas Aryeh ibid]
Do those who fast, also fast on the nights of R”H? No. [Piskeiy Hasiddur 115; Shaar Hakolel 34/11 based on Admur in Siddur regarding one who forgot Yaleh Veyavo on the night of R”H; Yifei Laleiv 2/5] However from other Poskim it is evident that they would fast even on the night of R”H. [See Admur 188/10; M”A 188/7; Kaf Hachaim 597/7] See Admur 597/5 that states regarding one who is fasting a Taanis Chalom on R”H “However it is forbidden to fast on the two night of R”H just like every other Yom Tov..” Veztaruch Iyun if this is going according to the final ruling in 597/1 that one may not fast or even according to the opinion that says one is to fast.
Do those that have fasted one time in the past have to fast every Rosh Hashanah? There are communities that have a tradition that whoever is accustomed to fast on R”H and does not fast one year will not complete his year in this world. [Admur 597/2; Michaber 597/2; Maggid Meisharim brought in Alef Hamagen 597/10] Nevertheless one who is not afraid is not required to fast all of his days and is rather to simply perform Hataras Nedarim like any other Neder. [Admur ibid; Rama ibid; Kol Bo; M”A 597/2 that Maharam Lunza did not fast the next year and nothing evil happened to him; See Alef Hamagen ibid; Ashel Avraham Butchach 597; Chasam Sofer 168] Thus if he fasted three times or one time with intention to fast every year he is to do Hataras Nedarim. [M”B 597/6; M”E 597/8; Derech Hachaim 2]
If one accepted to fast on R”H but did not make a vow, is he obligated to fast? Yes as there are opinions that say he accepted upon himself to do a Mitzvah. [Admur 597/8; M”A 597/8]
Matters related to those that fast on R”H: 1) Must those that fast spend their entire day learning Torah and Davening? Yes. [597/3; M”A 597/1; M”B 597/3; unlike Darkei Moshe 529/2; See Kaf Hachaim 597/12] 2) May those that fast cook food for others on R”H? No. [597/4] However a wife may cook for her husband even if she is fasting. [M”B 597/3; M”E 597/6; Kaf Hachaim 597/13] 3) Do those that fast recite Aneinu on Rosh Hashanah? No. This is similar to Yom Kippur of which Aneinu is not recited as these are days of judgment and it is thus not applicable to say “Beyom Tzom Taaniseinu”. [597/9; Rama 597/3; M”A 597/8; M”B 597/13] Fasting on Shabbos R”H: See Poskim in Kaf Hachaim 597/15; Piskeiy Teshuvos 597/2; Sdei Chemed Mareches R”H 2/7 [dispute between Lev Chaim and Yifei Laleiv]; Fasting if one is a Baal Bris: See Poskim in Kaf Hachaim 597/16; Being Matir Neder of fasting: See Poskim in Kaf Hachaim 597/17; Fasting on day of Yartzite on R”H: See M”E 597/15.
Taanis Chalom: If one experienced a bad dream, but is in doubt if it is one of the dreams that one fasts for on Shabbos, then he may not fast. [Admur ibid; M”A 597/4] If however one is allowed to fast for the dream on Shabbos then certainly he may fast on Rosh Hashanah. [Kaf Hachaim 597/5 and so is evident from Admur 597/5-7]
If one fasted a Taanis Chalom on R”H must he fast every R”H from now on? Some have a tradition that one must fast from now on, on both days of R”H, if he fasted on the first day. [Hagahos Maimanis brought in Admur 597/5 and so rules Michaber 597/3] See Admur ibid for the reason. Practically one who does not desire to follow this tradition is not required to do so. [Admur 597/6; M”A 597/5 and 597/2 that Maharam Lunza did not fast the next year and nothing evil happened to him; M”E 597/8; M”B 597;3]
If one fasted a Taanis Chalom on R”H must he fast on a weekday as a Tikkun? No, as there are opinions that say one is even initially to fast on R”H. [Admur 597/7; Rama 597/3; Terumos Hadeshen; M”A 597/6; See Poskim in Kaf Hachaim 597/33]
 Admur ibid, as explained in 583/4 based on a verse in Nechemia
The reason: The Jews receive atonement when they eat drink and are merry. [Midrash Esther brought in Yifei Laleiv 3/1; Kaf Hachaim 597/2] Alternatively it is done in order so we have strength to fast on Tzom Gedalia. [Yifei Laleiv ibid]
 583/4; Rama 583/1 [“The custom is to eat fatty meat and all types of sweets”]
 Admur and Rama ibid explicitly mention fatty meat; Seemingly the novelty is that although on every Yom Tov one is to eat meat as part of the Mitzvah of Oneg [see 243/1 and 529] nevertheless on R”H one is to enhance this meat as it is to be fatty meat.
Other Opinions: The Beis Yosef writes in the Sefer Maggid Meisharim Netzavim in the name of his angelic teacher, the Maggid, that one is to avoid eating meat on R”H. Now although it states in Nechemia that one is to eat fatty foods, that is referring to the general populace, however the spiritually unique individual is to avoid doing so. Furthermore the term “fatty” used in Nechemia refers not to meat but to butter and dairy fat. [ibid brought in M”A 597/1] To note however that the M”A 597 writes in the name of the Maggid Meisharim that one is not to eat meat on R”H and does not differentiate between individuals, hence implying that this is a directive for all. [See Sdei Chemed Mareches R”H 2]. Practically we do not rule like this opinion, even if one is a spiritually upright individual, and on the contrary he is to be particular to eat meat as writes the Rama and Admur ibid, as well as all the other Achronim. [Yifei Laleiv 2/1; Kaf Hachaim 597/11; Sdei Chemed Mareches R”H 2]
 The Reason: As is written in Ezra regarding Rosh Hashanah “Eat fatty [foods] and drink sweet beverages”. [ibid]
 Ben Ish Chaiy Netzavim 4; Kaf Hachaim 583/24
 597/1; Michaber 597/1; Maggid Meisharim Netzavim “One is to diminish in other foods and drinks as well”
 529/6 [There is no obligation of Simcha on Rosh Hashanah unlike Yom Tov and Chol Hamoed] 582/10 [We omit Moadim Lesimcha] However see 597/1 “There is a Mitzvah to eat drink and rejoice”. Seemingly however one must conclude that this is not the obligated form of Simcha referred to in 529/6 and is rather a mere feeling of rejoicing.] See M”E 582/2 [regarding not saying Simcha Ubetzahala on R”H that falls on Shabbos]; Rambam Yom Tov 6/27
Other opinions: Many Poskim rule that the Mitzvah of Simcha is applicable equally to Rosh Hashanah. [M”B 597/1; Kaf Hachaim 597/1; Shaagas Aryeh 102; Yifei Laleiv 2/1; Teshuvos Mahril 128; Noda Beyehuda Kama 32; Chasam Sofer 168; See Sdei Chemed Mareches R”H 2] The reason for this is because Rosh Hashanah is included within the term Chag, as the verse states “Bekeseh Leyom Chageinu” and the verse states Vesamachta Bechagecha. Likewise the verse in Nechemia states that one needs to eat fat meats and sweet beverages. [M”B ibid; Kaf Hachaim ibid] See Sdei Chemed ibid for a full analysis on this subject.
 529/6 as explained in the previous Q&A!
Other opinions: See previous footnote.
 M”A 597/1 in name of Maggid Meisharim
The Beis Yosef writes in the Sefer Maggid Meisharim Netzavim in the name of his angelic teacher, the Maggid, that one is to avoid drinking wine on R”H. Now although it states in Nechemia that one is to drink sweet beverages, that is referring to the general populace, however the spiritually unique individual is to avoid doing so. Furthermore that is referring to non-alcoholic beverages that are sweet. [ibid] To note however that the M”A 597 writes in the name of the Maggid Meisharim that one is not to eat meat on R”H and does not differentiate between individuals, hence implying that this is a directive for all. [See Sdei Chemed Mareches R”H 2]
 We do not rule like this opinion, even if one is a spiritually upright individual and on the contrary he is to be particular to eat meat as writes the Rama and Admur ibid, as well as all the other Achronim. [Yifei Laleiv 2/1; Kaf Hachaim 597/11; Sdei Chemed Mareches R”H 2]
 In 583/4 Admur does not mention wine in his list of drinks. Now, the Magen Avraham 597/1 is of the opinion that one is not to drink wine or eat meat on Rosh Hashanah [other than for Kiddush]. Thus from the fact that Admur rules explicitly that meat is to be eaten [unlike the M”A] but does not make mention of wine, implies that perhaps one is not to drink wine as rules the M”A. Furthermore, Admur actually adds words to the wording of the Rama ibid and gives an example of a sweet drink “med” which perhaps is coming to exclude wine. On the other hand in 529/4 Admur clearly rules that one is to drink wine during the Yom Tov meals due to the Mitzvah of Oneg. In the next Halacha he then states that the Mitzvah of Oneg applies equally to Rosh Hashanah. Vetzaruch Iyun. Perhaps however one can say that Admur omitted wine being it is regardless an obligation of Oneg on R”H and hence there is no novelty in stating that due to Nechemia one needs to drink wine. [However regarding meat the novelty is that it is to be fatty] Hence in order to explain the novelty of Nechemia Admur explicitly mentioned a non-alcoholic sweet beverage to show that the obligation applies likewise towards it.
 Ateres Zekeinim 597; M”B 597/2; Kaf Hachaim 597/6
 Brought in Bach 591; M”A 591; See also Beis Yehuda p. 107; Sdei Chemed Mareches R”H 2