Let X and Y be topological spaces and C and D semigroups under composition of maps from X to X and Y to Y respectively. Let H be an isomorphism from C to D; it is shown that if both C and D contain the constant maps then there exists a bijection h from X to Y such that H(f) = h∘f∘h⁻¹, VfɛC. We investigate this situation and find sufficient conditions for this h to be a homeomorphism. In this regard we study the familiar semigroups of continuous, closed, and connected maps. An auxiliary problem is the case when C = D and H is an automorphism of D), We then ask when is every automorphism is inner. The question is answered for certain particular semigroups; e.g., the semigroup of differentiable maps on the reals has the property that all automorphisms are inner. / Science, Faculty of / Mathematics, Department of / Graduate