  We describe the analogue of the Sato-Tate conjecture for an abelian variety
over a number field; this predicts that the zeta functions of the reductions
over various finite fields, when properly normalized, have a limiting
distribution predicted by a certain group-theoretic construction related to
Hodge theory, Galois images, and endomorphisms. After making precise the
definition of the "Sato-Tate group" appearing in this conjecture, we describe
the classification of Sato-Tate groups of abelian surfaces due to
Fite-Kedlaya-Rotger-Sutherland. (These are notes from a three-lecture series
presented at the NATO Advanced Study Institute "Arithmetic of Hyperelliptic
Curves" held in Ohrid (Macedonia) August 25-September 5, 2014, and are expected
to appear in a proceedings volume.)
