  This is a survey on motivic zeta functions associated to abelian varieties
and Calabi-Yau varieties over a discretely valued field. We explain how they
are related to Denef and Loeser's motivic zeta function associated to a complex
hypersurface singularity and we investigate the relation between the poles of
the zeta function and the eigenvalues of the monodromy action on the tame
$\ell$-adic cohomology of the variety. The motivic zeta function allows to
generalize many interesting arithmetic invariants from abelian varieties to
Calabi-Yau varieties and to compute them explicitly on a model with strict
normal crossings.
