Let C be a fusion category faithfully graded by a finite group G and let D be
the trivial component of this grading. The center Z(C) of C is shown to be
canonically equivalent to a G-equivariantization of the relative center Z_D(C).
We use this result to obtain a criterion for C to be group-theoretical and
apply it to Tambara-Yamagami fusion categories. We also find several new series
of modular categories by analyzing the centers of Tambara-Yamagami categories.
Finally, we prove a general result about existence of zeroes in S-matrices of
weakly integral modular categories.