According to the Sato–Tate conjecture (now a theorem in many cases), if one chooses an elliptic curve over a number field and renormalizes the factors of its L-function, they exhibit one of exactly three possible distributions determined by the endomorphism ring of the curve (and in particular whether or not it has complex multiplication). The definition of the Sato–Tate group seeks to model the corresponding situation for curves of higher genus (or even other motives). In 2012, the Sato–Tate groups of abelian surfaces were classified by Fite–Rotger–Kedlaya–Sutherland; there are 52 of them. With Fite and Sutherland, we are currently classifying the Sato-Tate groups of abelian threefolds; come to the talk to find out how many there are!