In Eretz Yisrael, Shmini Atzeret and Simchat Torah are observed on the same day. Outside of Eretz Yisrael, Simchat Torah is the second day of what becomes a two-day festival.

We begin mentioning rain in the second paragraph of the musaf amida of Shmini Atzeret and continue until the first day of Passover.
 
The universal custom is to complete the reading of the Torah and to begin reading it again on Simchat Torah. Dancing and singing with the Torah scrolls is also a universal custom.