Path integrals are considered for the cases where the underlying manifold is multiply connected or non-flat. In case of multiple connectivity, the contributions of different homotopy classes of paths are analyzed with the help of covering spaces. In case of a non-flat manifold, it is pointed out that a judicious choice of the free Hamiltonian operator and of normalizing factors can eliminate the explicit occurrence of the scalar curvature. (A heuristic approach to path integrals is adopted in case of non-flat spaces.)