We construct the covariant path integral for the Neveu-Schwarz-Ramond superstring in superspace, with manifest invariance under diffeomorphisms and local supersymmetry transformations. Spin structure is introduced, and the constraints imposed by modular invariance on fermionic string models examined. The critical exponents of the bosonic and fermionic string models in space-time dimensions less than the critical dimension are obtained for world surfaces of arbitrary topology.