  In this contribution I show that it is possible to construct
three-dimensional spaces of non-constant curvature, i.e. three-dimensional
Darboux-spaces. Two-dimensional Darboux spaces have been introduced by Kalnins
et al., with a path integral approach by the present author. In comparison to
two dimensions, in three dimensions it is necessary to add a curvature term in
the Lagrangian in order that the quantum motion can be properly defined. Once
this is done, it turns out that in the two three-dimensional Darboux spaces,
which are discussed in this paper, the quantum motion is similar to the
two-dimensional case. In $\threedDI$ we find seven coordinate systems which
separate the Schr\"odinger equation. For the second space, $\threedDII$, all
coordinate systems of flat three-dimensional Euclidean space which separate the
Schr\"odinger equation also separate the Schr\"odinger equation in
$\threedDII$. I solve the path integral on $\threedDI$ in the $(u,v,w)$-system,
and on $\threedDII$ in the $(u,v,w)$-system and in spherical coordinates.
