  We explore the $n$-body problem, $n\geq 3,$ on a surface of revolution with a
general interaction depending on the pairwise geodesic distance. Using the
geometric methods of classical mechanics we determine a large set of
properties. In particular, we show that Saari's conjecture fails on surfaces of
revolution admitting a geodesic circle. We define homographic motions and,
using the discrete symmetries, prove that when the masses are equal, they form
an invariant manifold. On this manifold the dynamics are reducible to a
one-degree of freedom system. We also find that for attractive interactions,
regular $n$-gon shaped relative equilibria with trajectories located on
geodesic circles typically experience a pitchfork bifurcation. Some
applications are included.
