We will introduce two invariants called the arithmetic Tutte polynomial and the Tutte quasi-polynomial, and we will survey their several applications, including: - generalization of graph colorings and flows to cellular complexes of higher dimension; - combinatorial and topological invariants of toric arrangements; - Ehrhart polynomials of zonotopes; - Hilberts series of some modules related to the vector partition function. The talk is partially based on joint work of the speaker with Petter Brändén, Michele D'Adderio and Alex Fink.