  We study finite quotients of abelian varieties (fqav for short) i.e.
quotients of abelian varieties by finite groups. We show that Q-abelian
varieties (i.e. fqav's with $\mathbb Q$-linearly trivial canonical divisors)
are characterized by the existence of quasi\'etale polarized (or int-amplified)
endomorphisms. We show that every fqav has a finite quasi\'etale cover by the
product of an abelian variety and a $\mathbb Q$-Fano fqav. Using such
coverings, we give a characterization of $\mathbb Q$-Fano fqav's, and show that
$\mathbb Q$-Fano fqav's and Q-abelian varieties are ``building blocks'' of
general fqav's.
