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When is the Shofar to be blown? 
The Shofar may only be blown during the daytime of Rosh Hashanah. The entire day is valid for blowing Shofar.
From what time may one begin blowing Shofar? One may blow Shofar beginning from sunrise. If one blew before sunrise but after the beginning of dawn [Alos Hashachar] he has nevertheless fulfilled his obligation and is not required to hear Shofar a second time. [Furthermore in a time of need one may even initially be lenient.] If one began blowing right before Alos and the sound continued past Alos, that sound is invalid. This applies even if it contained the minimum required sound from after Alos.
Until what time may one blow Shofar? One may blow Shofar until Tzeis Hakochavim. [Regarding if a blessing is made when blowing past sunset-See Q&A]
Blowing within the first three hours of the day: If one is blowing Shofar on Rosh Hashanah without a Minyan in order to fulfill the Mitzvah, he is to beware to delay the blowing of the Shofar until three hours have passed from the beginning of the day. [Some Poskim write that in today’s times it is proper not to blow the Shofar even with a Minyan until after three hours have passed.]
One may blow Shofar from sunrise until Tzeis Hakochavim.
May one blow past sunset?
One may blow Shofar from after sunset until nightfall. After nightfall it is forbidden to blow. Seemingly this applies likewise on the 2nd day of Rosh Hashanah, that one may blow until nightfall.
Is a blessing recited? Some Poskim rule a blessing is not to be recited when blowing Shofar between sunset and Tzeis. Other Poskim rule that on the first day one may blow with a blessing between sunset and Tzeis. However on the second day one must blow without a blessing.
May one blow Shofar past sunset until nightfall if the second day of Rosh Hashanah is Erev Shabbos?
May one blow Shofar if he already accepted Shabbos on R”H that falls on Erev Shabbos?
If one accepted Shabbos early, prior to sunset, then it is permitted for him to blow Shofar without a blessing up until sunset. If another person who did not yet accept Shabbos is able to blow the Shofar on his behalf then he may do so even with a blessing. In a case that one who already accepted Shabbos is blowing he is not to blow more than three sets of Tashrat; Tashat; Tarat. This means that he is to blow exactly three sets of each set and not anymore.
 588/1; Michaber 588/1
 The reason: This is learned from the verse “Yom Teruah” that the Teruah must take place during the day and not night. [Admur ibid; Megillah 20b]
 Biblically, day begins from Alos Hashachar regarding all Halachic matters. Nevertheless the Sages decreed that one must delay blowing the Shofar until after sunrise being that not all people are expert in the definition of Alos and hence in order to be assure that it is day one must wait until sunrise. [Admur ibid; Rashi Megillah ibid]
 Kaf Hachaim 588/8 regarding one who is traveling based on Gemara Brachos 30a
 587/4; 588/2
The reason: The reason for this is because every blow must be valid in its entirety, from beginning to end, for it to be considered a Kosher sound. Hence since in this case the sound only became valid in its middle, it therefore is not considered a Kosher sound. [587/4]
 591/14; M”A 591/9; M”B 591/15; Kaf Hachaim 591/36
 The reason: The reason for this is because within the first three hours of the day G-D does not judge beyond the letter of the law, as during this time He learns His Torah which is called Truth [and thus judges people in accordance to Truth]. There is thus to worry that perhaps the supernal courts will investigate his actions and his prayers [Tekios] will be denied. However after three hours into the day He judges the entire world and once He sees that the world cannot up stand to His judgment He therefore judges them in a manner of above the letter of the law, and the individuals prayer will not be differed. [ibid]
 Machazik Bracha 588/1; Mateh Efraim 588/2; Kaf Hachaim 588/6
 Admur 588/1; Chayeh Adam 141/1; Mishneh Berurah 588/1; Kaf Hachaim 588/2.
 So is implied from Stimas Haposkim brought above. However Tzaruch Iyun as why do we not say Safek Derabanan Lehakeil, and hence there is no longer an obligation to blow by Bein Hashmashos which is questionably day/night. This would be similar to the ruling that on the 2nd day one who is in doubt as to whether he heard Shofar is not required to blow. However perhaps there is a difference if the doubt is in whether one already heard the actual blowing, or if one for certain did not hear the blowing but is in doubt if the Mitzvah still applies.
 Chayeh Adam 141/1; 144/20; Mishneh Berurah 588/1; Kaf Hachaim 588/2; 600/12
 Michtam Ledavid 17; Birkeiy Yosef 600/5, brought in Kaf Hachaim 600/11
Opinion of Admur: Admur rules that from Alos until Tzeis is the permitted time for blowing Shofar. The source of Admur if from the Gemara Megila 20b which states that until Tzeis Hakochavim it is considered day. This seems to imply that during Bein Hashmashos one may blow even with a blessing! This is further inferred from the change of wording from the Tur which writes simply “until night” while Admur specifies Tzeis. Vetzaruch Iyun Gadol, as Bein Hashmashos is questionable whether it is night or day as rules Admur is 261/1, hence how can one say a blessing at that time.
 Michtam Ledavid 17 [“certainly he is to blow”]; and so rule the following Poskim regarding reading the Megillah by Bein Hashmashos of Erev Shabbos: P”M 692 M”Z 3; Tzitz Hakodesh 55; Nitei Gavriel 31/10 footnote 15
 The reason: Although certainly one is exempt from the Mitzvah as it is a Safek Derabanan, nevertheless the Sages permit one to do Shvus in order to calm the person and remove anguish from his soul that he did not fulfill the Mitzvah. This is no different than the allowance to permit a woman to ask a gentile to light candles for her by Bein Hashmashos due to her anguish. [see 261/2; 263/11] Thus here too it is allowed, as he is no less than this woman as he desires to do a Mitzvah in order to relieve his anguish in which case the Sages allowed during Bein Hashmashos if the Melacha is only Rabbinical. [Michtam Ledavid ibid; Admur 342/1; 261/2] This allowance certainly should apply towards Shofar blowing which is not even a full Melacha but rather a mere Uvdin Dechol. [see Admur 588/4; 623/11 “not a Shvus Gamur”; 299/15; M”A 623/4; See also Michtam Ledavid ibid] One cannot however say that since Shofar is a Mitzvah and is merely a Shvus it thus is to be allowed, as is the rule with any Rabbinical prohibition which is allowed to be done for the sake of a Mitzvah during Bein Hashmashos, as perhaps we only allow a definite Mitzvah to be done by Bein Hashmashos. However if it is a questionable Mitzvah then perhaps we do not allow it to be done. This especially applies here in this case where the Mitzvah of blowing Shofar on the second day is only Rabbinical, and hence it is a doubt in a Rabbinical Mitzvah. On the other hand if one understands Admur to rule that the Mitzvah applies until Tzeis, then it is still a definite Mitzvah by Bein Hashmashos, and thus is allowed to be done. Vetzaruch Iyun!
 Mateh Efraim 601/13; M”B 600/7; Kaf Hachaim 600/13; Implication of Taz 600/1 [He differentiates between accepting Shabbos before Bein Hashmashos and Bein Hashmashos itself, saying that Bein Hashmashos is more severe. Likewise in the Teshuvah he always mentioned that the allowance is before sunset. However see Admur 261/3 that accepting Shabbos is more severe than Bein Hashmashos regarding doing Rabbinical Melacha.]
 The reason: As the second day is only Rabbinical and we hence apply the rule of Safek Derabanan Lekula towards the Mitzvah. [M”B ibid] Hence there is no Mitzvah at all to blow and therefore there is no reason to permit it. However see 261/2 that even a Yartzite candle may be lit by a gentile even though this does not involve a Mitzvah and hence the claim of the Poskim that permit it is valid. Another possible reason to forbid it is because it is known that the Sages forbade blowing Shofar on Shabbos Rosh Hashanah due to a decree that one may come to carry the Shofar [which is possibly a Biblical prohibition]. Hence perhaps by Bein Hashmashos the Sages likewise applied this decree. Furthermore, if when it is for certain a Mitzvah of Biblical nature the Sages decree that it not be done, then certainly when it is a Mitzvah of Rabbinical nature and a questionable one at that, the Sages would decree that it should not be done. On the other hand one can say that the Sages only made the decree to uproot the entire Mitzvah, however once the Mitzvah became active, as in our case on Friday then the decree no longer uproots it by Bein Hashmashos. Furthermore, if one says that Bein Hashmashos is night then there is no Mitzvah to blow Shofar in which case the Sages did not make a decree against carrying it. The decree against carrying it was only made when the Mitzvah applies. And if one says Bein Hashmashos is still day, then according to all one may still carry.
 Taz 600/1; Erech Hashulchan 600/2; Elya Raba 601; Ikarei Daat 30/24; M”E 601/13; Chayeh Adam 144/20; M”B 600/7
The reason: As one does not intend to accept Shabbos to the point that he can’t fulfill the Mitzvah of Shofar. Likewise if the Sages permit doing a Mitzvah during Bein Hashmashos then certainly they permit it before Bein Hashmashos, after he accepted Shabbos. However a blessing may not be made as this shows that he is doing so for the sake of the Mitzvah and it hence carries the decree of Raba that one should not blow Shofar for the Mitzvah on Shabbos. [Taz ibid; Vetzaruch Iyun as Admur rules in 261/3 that accepting Shabbos is more severe than Bein Hashmashos regarding doing Rabbinical Melacha; See Sefer Hachaim; Yalkut Gershoni 30; Mahrsham 3/333; Divrei Yisrael 165]
Other Opinions: Some Poskim rule that he may blow the Shofar even with a blessing. [Michtam Ledavid 17; Birkeiy Yosef 600/5, brought in Kaf Hachaim 600/11] Some Poskim rule one may not blow the Shofar all, even without a blessing. [Mor Uketzia 600; Hagahos Hatur brought in Kaf Hachaim 600/12]
 Mateh Efraim 601/13; M”B 600/7 in Shaareiy Tziyon 600/6; Kaf Hachaim 600/13
 M”E ibid; M”B ibid
 Kaf Hachaim 600/13