In this paper we extend categorically the notion of a finite nilpotent group
to fusion categories. To this end, we first analyze the trivial component of
the universal grading of a fusion category C, and then introduce the upper
central series of C. For fusion categories with commutative Grothendieck rings
(e.g., braided fusion categories) we also introduce the lower central series.
We study arithmetic and structural properties of nilpotent fusion categories,
and apply our theory to modular categories and to semisimple Hopf algebras. In
particular, we show that in the modular case the two central series are
centralizers of each other in the sense of M. Muger.