It is a well-known result of Dirichlet that if $n$ and $a$ are relatively prime integers, then the sequence $\{k n+a:k\geqslant 0\}$ contains infinitely many prime numbers. I will start by reinterpreting this result using the Chebotarev density theorem. Then I will move on to the "classical" Sato-Tate conjecture, and show how the Sato-Tate measure naturally arises from group theory. I will explain the relationship between equidistribution theorems and $L$-functions — in particular, how the analytic continuation and nonvanishing of a family of $L$-functions implies an equidistribution theorem. Finally, I will discuss Serre's "generalized Sato-Tate conjecture" and show that both the Chebotarev density theorem and the classical Sato-Tate conjecture are special cases.