A toric arrangement is a finite family of hypersurfaces in a torus, every hypersurface being the kernel of a character. We describe some properties of such arrangements, by comparing them with hyperplane arrangements. The Tutte polinomial is an invariant which encodes a rich description of the topology and the combinatorics of a hyperplane arrangement, and satisfies a simple recurrence. We introduce the analogue of this polynomial for a toric arrangement. Furthermore, we show that our polynomial computes the volume of the related zonotope, counts its integral points, and provides the graded dimension of a space of quasipolynomials introduced by Dahmen and Micchelli to study partition functions.