The Shabbat during the Ten Days of Repentance is called “Shabbat Shuva” or “Shabbat Teshuva” after its haftara, which begins, “Return (shuva), O Israel, to the Lord your God, for you have fallen because of your sin” (Hoshea 14:2). It is also known that Shabbat is the source of blessing, and everything that happens during the week draws its vitality from Shabbat. In a sense, then, this Shabbat can be considered the root of Yom Kippur, and thus it is proper to use it for repentance (teshuva) and Torah study.
There is a widespread custom for the rabbi to give a special sermon on this Shabbat, dealing with the relevant laws of the period and rebuking the community with the goal of inspiring them to repent from their common sins and strengthen their Torah study, charity, and mitzva observance. This sermon is one of the two most important sermons of the year (the other being that of Shabbat Ha-gadol before Pesaḥ). Therefore, even if other rabbis give lectures and sermons during the year, on this Shabbat the local rabbi should speak. Since it is his community, he knows best which issues need to be addressed and what needs to be improved. Everyone should attend this sermon, from Torah scholars to simple and uneducated people who will not fully understand the rabbi. Some rabbis wrap themselves in a talit when they speak, in honor of the audience and the sermon. (See Mateh Ephraim 602:41; Elef Ha-magen ad loc. 23.)
In the past, some communities held the sermon in the morning, between Shaḥarit and the Torah reading. In large communities with many synagogues, where it is difficult to gather everyone together in the middle of the service, the sermon is held right after Minḥa. Minḥa is scheduled for two or three hours before shki’a so the rabbi can speak at length and still leave time to eat se’uda shlishit (Mateh Ephraim 602:42). In many communities, the sermon is delivered between se’uda shlishit and Ma’ariv.
On Shabbat Shuva (and the preceding Friday afternoon), Ashkenazim, Yemenites, and some Sephardim (including many of North African descent) do not recite Avinu Malkeinu, since Shabbat is not a time for personal requests. Even if Rosh Ha-shana itself is on Friday or Shabbat, they do not recite Avinu Malkeinu (Sefer Ha-pardes; Roke’aḥ; Ran; Responsa Rivash §512; Rema 602:1; MB 584:4). Many Sephardim do say Avinu Malkeinu on Shabbat during the Ten Days of Repentance, including Rosh Ha-shana. This does not show disrespect for Shabbat, since these days are designated for repentance and prayer for the upcoming year (R. Amram Gaon; Rabbeinu Gershom; Me’iri; Kaf Ha-ḥayim 584:7-8).
Many people wait to recite Kiddush Levana until after Yom Kippur, because Kiddush Levana should be recited joyfully, and only after Yom Kippur does the tension of judgment dissipate. Then people leave the synagogue feeling joyful (Maharil; Rema 602:1; Pri Ḥadash; R. Mordechai Eliyahu; 7:19 below). Others disagree, maintaining that people should recite the prayer during the Ten Days of Repentance, as doing so might tip the scales in their favor (Levush; Ḥida; Vilna Gaon). Every community should continue with its custom.