The anti-Ramsey number of Erdös, Simonovits and Sós from 1973 has become a classic invariant in Graph Theory. To extend this invariant to Matroid Theory, we use the heterochromatic number hc(H) of a non-empty hypergraph H. The heterochromatic number of H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a totally multicoloured hyperedge of H.Given a matroid M, there are several hypergraphs over the ground set of M we can consider, for example, C(M), whose hyperedges are the circuits of M, or B(M), whose hyperedges are the bases of M. We determine hc(C(M)) for general matroids and characterise the matroids with the property that hc(B(M)) equals the rank of the matroid. We also consider the case when the hyperedges are the Hamiltonian circuits of the matroid. Finally, we extend the known result about the anti-Ramsey number of 3-cycles in complete graphs to the heterochromatic number of 3-circuits in projective geometries over finite fields, and we propose a problem very similar to the famous problem on the anti-Ramsey number of the p-cycles.