This extended abstract is an overview of the results of my recent paper. We present two general sets of functionals for which parabolic phase space Feynman path integrals on the torus have a mathematically rigorous interpretation. More precisely, for each functional belonging to each set, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the ending point of position paths and to the starting point of momentum paths. Each set of functionals is closed under addition, multiplication, translation, invertible integer linear transformation, and functional differentiation. Consequently, we can construct a large number of path-integrable functionals. While we must exercise caution when using the phase space path integrals, several properties analogous to those of conventional integrals are valid in the phase space path integrals.