A general complementarity problem with respect to a convex cone and its polar in a locally convex, vector-topological space is defined. It is observed that, in this general setting, the problem is equivalent to a variational inequality over a convex cone. An existence theorem is established for this general case, from which several of the known results for the finite-dimensional cases follow under weaker assumptions than have been required previously. In particular, it is shown that, if the given map under consideration is strongly copositive with respect to the underlying convex cone, then the complementarity problem has a solution.