  We consider the $N$-body problem of celestial mechanics in spaces of nonzero
constant curvature. Using the concept of locked inertia tensor, we compute the
moment of inertia for systems moving on spheres and hyperbolic spheres and show
that we can recover the classical definition in the Euclidean case. After
proving some criteria for the existence of relative equilibria, we find a
natural way to define the concept of central configuration in curved spaces
using the moment of inertia, and show that our definition is formally similar
to the one that governs the classical problem. The existence criteria we
develop for central configurations help us provide several examples and prove
that, for any given point masses on spheres and hyperbolic spheres, central
configurations always exist. We end our paper with results concerning the
number of central configurations that lie on the same geodesic, thus extending
the celebrated theorem of Moulton to hyperbolic spheres and pointing out that
it has no straightforward generalization to spheres, where the count gets
complicated even in the case $N=2$.
