If a group G operates on the spaces X and Y we define a G-equivariant map f: xY to be one having the property fg = gf, all g e G. In the first part of this paper we give a necessary condition for the existence of equivariant maps and examine some of its consequences. In the second we show that under appropriate circumstances the condition is sufficient as well as necessary. Our method provides, in fact, an equivalence of the class of all maps of one space into another equivariant with respect to a group of operators, with another class of maps-the right inverses of a "collapsed projection." We give an application to the definition of tensor functions on manifolds.