  In this paper path integration in two- and three-dimensional spaces of
constant curvature is discussed: i.e.\ the flat spaces $\bbbr^2$ and $\bbbr^3$,
the two- and three-dimensional sphere and the two- and three dimensional
pseudosphere. The Laplace operator in these spaces admits separation of
variables in various coordinate systems. In all these coordinate systems the
path integral formulation will be stated, however in most of them an explicit
solution in terms of the spectral expansion can be given only on a formal
level. What can be stated in all cases, are the propagator and the
corresponding Green function, respectively, depending on the invariant distance
which is a coordinate independent quantity. This property gives rise to
numerous identities connecting the corresponding path integral representations
and propagators in various coordinate systems with each other.
