SUNSET WEDNESDAY, SEPTEMBER 25 and SUNSET THURSDAY, SEPTEMBER 26: The joyful days of eating (and for some nature-loving families actually sleeping) in the sukkah are ending. But, some extend eating in the sukkah just one more day—and today Jews celebrate Shemini Atzeret and then the closely connected Simchat Torah. For Jews in Israel, these two holidays combine into one day; for Jews of the Diaspora, Shemini Atzeret is followed by Simchat Torah by one day.
Distinct from Sukkot, when rain may be unpleasant inside the loosely thatched sukkah, Jews begin praying for rain on Shemini Atzeret. The rainy season in Israel begins soon, and for agricultural purposes, the Musaf Amidah prayer is recited, for rain, on Shemini Atzeret.
Great happiness continues on Simchat Torah as synagogues around the world hold processions; joyful dancing and singing ensue. Torah scrolls are carried through the aisles, and even children join in by carrying toy or paper versions of the scrolls, making their way around the building in a series of seven circuits (hakafot). The primary celebration of Simchat Torah begins in the evening. When the ark is opened, congregation members get up to sing and dance. In many regions, the singing and dancing is taken to the streets and lasts many hours. (Wikipedia has details.)
FOR MORE ON THESE HOLIDAYS: Enjoy this introduction by Debra Darvick, author of This Jewish Life.