  The two-body problem with a central interaction on simply connected constant
curvature spaces of an arbitrary dimension is considered. The explicit
expression for the quantum two-body Hamiltonian via a radial differential
operator and generators of the isometry group is found. We construct a
self-adjoint extension of this Hamiltonian. Some its exact spectral series are
calculated for several potential in the space ${\bf S}^3$.
  We describe also the reduced classical mechanical system on a homogeneous
space of a Lie group in terms of the coadjoint action of this group. Using this
approach the description of the reduced classical two-body problem on constant
curvature spaces is given.
