We introduce a trivariate polynomial invariant for signed graphs, which we call the signed Tutte polynomial, and show that it contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the signed Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the "dichromate" for signed graphs (our signed Tutte polynomial) differs from the dichromatic polynomial (the rank-size generating function). The signed Tutte polynomial is a special case of a trivariate "joint Tutte polynomial" of ordered pairs of matroids on a common ground set -- for a signed graph, the cycle matroid of its underlying graph and its signed-graphic matroid form the relevant pair of matroids. This is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and contains as a specialization the Tutte polynomial of a matroid perspective defined by Las Vergnas.