In this paper the Feynman path-integral technique is applied for superintegrable potentials on two-dimensional spaces of nonconstant curvature: these spaces are Darboux spaces D_I and D_II, respectively. On D_I there are three and on D_II four such potentials, respectively. We are able to evaluate the path integral in the most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave functions, and the discrete energy spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D_I, in particular, show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant zero curvature) and the two-dimensional hyperboloid (constant negative curvature) emerge.

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