We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group over acting on an affine -variety , we consider the induced dual action on the coordinate ring and on the linear dual space of . In this setting, given an invariant closed semialgebraic subset of , we study the problem of representation of invariant nonnegative polynomials on by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on by invariant measures supported on . To this end, we analyse the relation between quadratic modules of and associated quadratic modules of the (finitely generated) subring of invariant polynomials. We apply our results to investigate the finite solvability of an equivariant version of the multidimensional -moment problem. Most of our results are specific to the case where the group is compact.