the (general) sato-tate conjecture for an abelian variety a of dimension g defined over a number field k predicts the existence of a compact subgroup st(a) of the unitary symplectic group usp(2g) that is supposed to govern the limiting distribution of the normalized euler factors of a at the primes where it has good reduction. for the case g=1, there are 3 possibilities for st(a) (only 2 of which occur for k=q). in this talk, i will give a precise statement of the sato-tate conjecture for the case of abelian surfaces, by showing that if g=2, then st(a) is limited to a list of 52 possibilities, exactly 34 of which can occur if k=q. moreover, i will provide a characterization of st(a) in terms of the galois-module structure of the r-algebra of endomorphisms of a defined over a galois closure of k. this is a joint work with k. s. kedlaya, v. rotger, and a. v. sutherland