[en] The conditions for two-dimensional (p, q) supersymmetric non-linear sigma models with Wess-Zumino term to have rigid 'isometry' symmetries are derived and related to the target-space geometry. This paper addresses the gauging of such rigid symmetries by coupling to (p, q) super Yang-Mills multiplets. We formulate (p, q) super Yang-Mills theory in (p, q) superspace and find that, in addition to the conventional multiplets, there are new twisted Yang-Mills multiplets. The (p, 0) and (p, 1) supersymmetric sigma models are coupled to (p, 0) and (p, 1) super Yang-Mills to obtain gauge-invariant actions in (p, 0) and (p, 1) superspace, respectively. The (2, 2) sigma models with torsion and off-shell closure of the supersymmetry algebra can be formulated in terms of chiral and twisted chiral (2, 2) superfields and we find that the gauging of such models requires both conventional and twisted (2, 2) Yang-Mills multiplets. We give two versions of the gauged (2, 2) sigma model with torsion. The (4, q) gauged supersymmetric models are finite by power counting. (orig.)