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Upon seeing the moon at the beginning of the month one says a blessing called Birkat HaLevana (for the details and text see below). What's the significance of this Bracha? The Gemara Sanhedrin 42a says that saying Birkat HaLavana is like greeting the Shechina. Beiur Halacha 426:2 explains that by seeing the moon fulfill its mission consistently we recognize Hashem’s might and control over nature. He adds that to emphasize that we only pray to Hashem and the moon is just a tool by which we can recognize Hashem’s strength we say Aleinu afterwards. (See note for more about the significance of this Bracha.) 
- One is obligated to make the bracha of Kiddish Levana which begins Baruch Atta Hashem... Asher BeMaamaro Bara Shechakim.  The bracha is made while standing.
- The bracha is made immediately upon seeing the moon and during the bracha and afterwards some say that one shouldn't look at the moon.
- For the Ashkenazic text click here (from Tefilah.org). For the Sephardic text click here (from Tefilah.org).
When it should be said
The Molad for Sivan 5781 is 8:30pm Jerusalem time on Tuesday 11th May 2021 and the latest time for kiddush levana for Sephardim is 8:30pm Jerusalem time on Wednesday 26th May 2021 and for Ashkenazim 2:52pm Jerusalem time on Wednesday 26th May 2021.
- According to Ashkenazim, one may say the Bracha 3 days after the molad, however according to Sephardim, in general one should wait until 7 days after the molad. For this halacha, days are counted by 24 hour periods.  It’s proper to wait until Motzei Shabbat (Saturday night) to say Birkat HaLevana. 
- During the month of Tishrei, Kiddish Levana isn't said prior to Yom Kippur. Rather it should be said right after Yom Kippur.
- During the month of Av, most have the minhag not to say Kiddush Levana prior to Tisha BeAv. Some say that it shouldn't be said right after Tisha BeAv, but in the next few days. 
- One may say Birkat HaLevana until 15 days after the molad, however, on the 16th day one should say it without Shem UMalchut. Some say that one may only say it until 14¾ days after the molad.  A chart of the time of the upcoming molads is printed here (see note for calculation):
- The strict halacha allows saying birkat halevana on Shabbat and Yom Tov, but since the kabbalists recommend against it one shouldn't do it unless one is running out of time to say it. 
Where it should be said
- Preferably, kiddush levana should be said outside. However, if one is not feeling well or it is dirty outside (to the extent that he would not be able to make a bracha in that place) he can say it inside. 
- In a situation where he needs to say it inside it is best to open the window or door, but if this too is not possible (e.g. the person finds it too cold) he may say it inside without even opening the window or door (taking into account he is sure that the light he sees is from the moon). 
- Although it seems that the custom of many is to make sure to say kiddush levana not standing under anything but the sky, according to the strict halacha it seems that one is permitted to say kiddush levana standing under a tree or a roof, as long as he exited the building. 
Looking at the moon
- One should look at the moon prior to saying the Bracha. Some hold that one should only look at the moon once and not look at it again, whereas others are lenient to allow looking at it anytime during the Seder of Birkat HaLevana except during the Bracha itself. 
- If one made Birkat HaLevana without looking at the moon but said it as part of the tzibbur, one fulfills their obligation bedieved. 
Saying Amen Amen
- When saying Amen Amen Amen in Birkat HaLevana one should make sure to pause between each Amen unless one is saying it in private in which case one doesn’t need to pause. 
- Women are exempt from Birkat HaLevanah. Even though Ashkenazic women usually are permitted to volunteer to make a bracha even when they are exempt, the minhag is that they don't say Birkat HaLevanah.  Others argue that they should really recite it.
- Blind men should recite birkat halevana even though they cannot see the moon. 
- Ashkenazic text of Birchat Halevana: Wikisource.org.
- Sephardic text of Birchat Halevana: Sefaria.org.
- The minhag is to say Shalom Aleichem three times during Birchat Halevana.
- Why did Chazal choose to make this Bracha on the moon specifically? Aruch HaShulchan 426:2 explains that since the moon is the closest to Earth we are able to recognize Hashem’s control over nature easily. Also, the moon is compared to Bnei Yisrael; just as the moon doesn’t have any of its own light, so too Bnei Yisrael are only sustained through Hashem and his Torah; just as the moon gets smaller and bigger, so too Bnei Yisrael are always renewed.
- Shulchan Aruch 426:1
- Gemara Sanhedrin 42a, Rama 426:1
- Mishna Brurah 425:13 citing the Magen Avraham
- Shulchan Aruch 426:4 writes that one shouldn’t say Birkat Halevana until 7 days have passed from the molad based on the Shaarei Orah. However, the Bach 426 rules like the Rabbenu Yonah who says that one may say it after 3 days. Taz 426:3, Magen Avraham 426:13, Aruch HaShulchan 426:13, and Mishna Brurah 426:20 agree with the Bach. Sh”t Yabia Omer 6:38(1) rules like Shulchan Aruch but adds that if Motzei Shabbat is a few hours less than 7 days one may say it and in places where majority of the month is cloudy one may say it after 3 days. Mishna Brurah 426:17 clarifies that for this halacha days are counted as 24 hour periods and not from sunset to sunset.
- Masechet Sofrim 19:10 (in some versions 20:1) writes that one should say Birkat HaLevana specifically on Motzei Shabbat. However, Rabbenu Yonah (Brachot 21a) quotes some who say this, rejects it, and concludes that one can say it anytime after 3 days. Bet Yosef 426:2 writes that probably Rabbenu Yonah didn’t have this girsa in Maasechet Sofrim. S”A 426:2 rules that one should say Birkat HaLevana on Motzei Shabbat. However, the Biur HaGra 426:2 cites the Gemara Yevamot 39a which says that we don’t delay performing Mitzvot. The Mishna Brurah 426:20 rules that it’s proper to wait until Motzei Shabbat but there is what to rely on to say it earlier and in the winter and rainy months those who are zealous to say it early are praiseworthy. Rama 426:2 writes that if Motzei Shabbat is the 11th night from the molad one doesn’t have to wait until Motzei Shabbat because it’s possible it will be cloudy the following 4 nights.
- Rama 426:2. The Beiur Halacha (sham d"h v'lo kodem), however, does quote the opinion of the Levush that by Yom Kippur, as opposed to Tisha B'Av, kiddush levana SHOULD be said BEFORE Yom Kippur as even one merit (one mitzvah) may tip the scales towards the positive during this period of judgement. He notes that this seems to be the opinion of the Beis Meir as well
- Rama 426:2, 551:8
- Rama 426:2, Mishna Brurah 426:10. However, Ish Matzliach footnote 3 on the Mishna Brura says that the minhag is most places is to say it on Motzaei Tisha B'av
- Shulchan Aruch 426:3 rules like the opinion of Nehardai in Gemara Sanhedrin 41b who say that one has 15 complete days. Chazon Ovadia Chanuka p. 349 agrees unlike the opinion of the Rav Poalim. However, the Rama 426:3 writes that one may only say it up to half of 29 days, 12 hours, and 793 chalakim. Beiur Halacha s.v. VeLo writes that if one is in such a situation where half of 29 days, 12 hours, 793 chalakim passed and 15 days have not, there is what to rely on to say the Bracha. Beiur Halacha adds that if it’s the 16th day one should say it without Shem UMalchut. Sh”t Yabia Omer 6:38 agrees with the Beiur Halacha that on the 16th day one should say it without Shem UMalchut.
- Rambam (Kiddush HaChodesh 6:3) writes that from one Molad (the astronomical time when the new moon can be seen) to another is 29 days, 12 hours, and 793 chelakim (and there's 1080 chelakim in an hour). The Rambam 6:8 also writes that the time with which to begin to calculate the Molads from the beginning of creation is the 2nd day, 5 hours, and 204 chelakim. To calculate the Molad of Rosh Chodesh Tishrei 5773, first let us calculate the number of months which passed from creation and then add the appropriate time for each month. Although there are 12 months in a regular year because there are 7 leap years in a 19 year cycle, there is an average of 12.3684 lunar months a year. By Tishrei 5773, 5772 complete years have passed, meaning that 71390 complete months have passed since creation. The chelakim for the molad of Tishrei 5773 should be 1034 (i.e. 204 + (71390 * 793) modulo 1080) which is equal to 57 minutes and 8 chelakim. Making similar calculations leads to conclude that the Molad for Tishrei 5773 is 1am (i.e. 5+12*71390+(204+71930*793-1034)/1080) mod 24 starting from 6pm) Sunday (i.e. 2+1*71390+(909103-37879)/24 mod 7). The other months can then be calculated from that starting point by either adding or subtracting the interval for each month (1 day, 12 hours, 793 chelakim. To confirm these calculations, one can check the Chabad.org page of Molad times.
- The Rashba (responsa 4:48) quotes someone who said that it is forbidden to say birkat halevana on Friday night since it is like travelling out of the techum above ten tefachim. The Rashba doesn't understand that opinion. He argues that there is no techum above ten tefachim and also it is greeting Hashem and not the moon; therefore, it isn't at all like travelling out of the techum and is permissible to be recited on Shabbat. The Mishna Brurah 426:12 rules that it is permissible to say kiddush levana on Shabbat and Yom Tov if it is necessary. In the Shaar Hatziyun 426:12 he adds a reason that everyone can agree with; since it is a mitzvah filled with happiness a person might come to dance and it is forbidden to dance on Shabbat.
- Mishnah Brurah 426:21
- Shaar Hatziyun 426:25
- Rama 426:4 writes that "we do not sanctify the moon under a roof." The Mishnah Brurah 426:21 explains that since sanctifying the moon is like greeting the presence of the Shechinah (Sanhedrin 42a), it is not honorable to stand under a roof, rather we exit from under the roof to the street just like we would to greet a flesh and blood king. However, both the Chazon Ish (Orchot Rabbenu v. 1 p. 178) and Rav Chaim Kanievsky (Ishei Yisrael 40:15) say that the main idea is to leave one's house or shul, but once one has done that he can certainly say kiddush levana under a tree or overhang.
- Shulchan Aruch 426:2 writes that one should look at the moon prior to making the Bracha. Magen Avraham 426:8 quotes the Shlah who says that one should only look once and then it’s forbidden to look again (for Kabbalistic reasons). Shaarei Knesset HaGedolah 426:5 quotes the Sefer Charedim that one may only look at the moon at the time of the Bracha.
- However, Shaarei Knesset HaGedolah writes that the minhag is look at the moon during the entire Seder and it’s supported by the language of Maasechet Sofrim quoted by the Tur and S”A. Birur Halacha (vol 5, pg 122) writes that if one is looking at the moon in order to recognize Hashem’s wonders then it’s permissible.
- Mishna Brurah 426:13 quotes these three opinions and doesn’t give a ruling. The Aruch HaShulchan 426:5 writes that the minhag is like the Shlah. The Ben Ish Chai (Vayikra(2) 23), Kaf HaChaim Palagi 35:4, Kaf HaChaim Sofer 426:34, Mekor Chaim 426:2, Chesed Alafim 426:4, Yosef Ometz 474, Chida in Moreh Etzbah 6:186, Birkat Eitan (Birkat Levana pg 207), Yalkut Yosef (Birkat Levana pg 146), and Ateret Paz (Birkat Levana pg 65) agree.
- Birkat Eitan (Birkat Levana pg 200), Sefer Kiddish Levana (2 note 20) in name of Rav Elyashiv, Halichot Shlomo 15:13, Even Yisrael 426:1, Sh"t Shevet HaLevi Y"D 5:125(4)
- Yerushalmi (Megillah 4:10) says that one shouldn’t say Amen Amen just like one shouldn’t say Shema Shema. This is quoted by the Ohel Moed (Kriyat Shema 1:7). Bet Yosef 61 writes that he doesn’t know what the source of the Ohel Moed is and challenges this because there are pesukim with the words Amen VeAmen.
- Pri Chadash 61:12 answers that in the pesukim it’s permitted if one pauses between the two words Amen. Korban HaEdah (Megillah 4:10) writes that in a congregation it’s forbidden but the pesukim were said in private and then it’s permitted. Kiseh Eliyahu 61:3 and Yabia Omer 10:5 agree. Yaavetz (Mor UKesiyah 61) argues that the Yershalmi is against the Bavli it is totally permissible to say Amen Amen.
- Yabia Omer 10:5 concludes that when saying Amen Amen Amen in Birkat Halevana one should pause between each Amen. This is also the ruling of Kaf HaChaim 61:43.
- Sh"t Ateret Paz (vol 4 Chap 10 #37), Magen Avraham 426:1, Sh"t Haelef Licha Shlomo OC 193
- Rav Soloveitchik (cited in Nefesh Harav pg. 176)
- Mishneh Brura 426:1
- Rav Mashash in Mayim Chayim 92 adds another reason. We are asking for Shalom for Klal Yisrael, between the parts of Hashem's name, and that the moon should return to its original completeness.