  We consider the two-body problem on surfaces of constant non-zero curvature
and classify the relative equilibria and their stability. On the hyperbolic
plane, for each q>0 we show there are two relative equilibria where the masses
are separated by a distance q. One of these is geometrically of elliptic type
and the other of hyperbolic type. The hyperbolic ones are always unstable,
while the elliptic ones are stable when sufficiently close, but unstable when
far apart.
  On the sphere of positive curvature, if the masses are different, there is a
unique relative equilibrium (RE) for every angular separation except {\pi}/2.
When the angle is acute, the RE is elliptic, and when it is obtuse the RE can
be either elliptic or linearly unstable. We show using a KAM argument that the
acute ones are almost always nonlinearly stable. If the masses are equal there
are two families of relative equilibria: one where the masses are at equal
angles with the axis of rotation (`isosceles RE') and the other when the two
masses subtend a right angle at the centre of the sphere. The isosceles RE are
elliptic if the angle subtended by the particles is acute and is unstable if it
is obtuse. At {\pi}/2, the two families meet and a pitchfork bifurcation takes
place. Right-angled RE are elliptic away from the bifurcation point.
  In each of the two geometric settings, we use a global reduction to eliminate
the group of symmetries and analyse the resulting reduced equations which live
on a 5-dimensional phase space and possess one Casimir function.
