  In this paper the Feynman path integral technique is applied to
two-dimensional spaces of non-constant curvature: these spaces are called
Darboux spaces $\DI$--$\DIV$. We start each consideration in terms of the
metric and then analyze the quantum theory in the separable coordinate systems.
The path integral in each case is formulated and then solved in the majority of
cases, the exceptions being quartic oscillators where no closed solution is
known. The required ingredients are the path integral solutions of the linear
potential, the harmonic oscillator, the radial harmonic oscillator, the
modified P\"oschl--Teller potential, and for spheroidal wave-functions,
respectively. The basic path integral solutions, which appear here in a
complicated way, have been developed in recent work and are known. The final
solutions are represented in terms of the corresponding Green's functions and
the expansions into the wave-functions, respectively. We also sketch some
limiting cases of the Darboux spaces, where spaces of constant negative and
zero curvature emerge.
