  We study products of irreducible theta divisors from two points of view. On
the one hand, we characterize them as normal subvarieties of abelian varieties
such that a desingularization has holomorphic Euler characteristic 1. On the
other hand, we identify them up to birational equivalence among all varieties
of maximal Albanese dimension. We also describe the structure of maximal
Albanese dimension varieties X with holomorphic Euler characteristic 1 and
irregularity 2dim X-1.
