For an abelian surface A over a number eld k, we study the limit-
ing distribution of the normalized Euler factors of the L-function of A.
This distribution is expected to correspond to taking characteristic poly-
nomials of a uniform random matrix in some closed subgroup of USp(4);
this Sato-Tate group may be obtained from the Galois action on any Tate
module of A. We show that the Sato-Tate group is limited to a particular
list of 55 groups up to conjugacy. We then classify A according to the
Galois module structure on the R-algebra generated by endomorphisms of
AQ (the Galois type), and establish a matching with the classi cation of
Sato-Tate groups; this shows that there are at most 52 groups up to con-
jugacy which occur as Sato-Tate groups for suitable A and k, of which 34
can occur for k = Q. Finally, we exhibit examples of Jacobians of hyperel-
liptic curves exhibiting each Galois type (over Q whenever possible), and
observe numerical agreement with the expected Sato-Tate distribution by
comparing moment statistics.