The theory of point vortices in a two-dimensional ideal fluid has a long history, but on
surfaces other than the plane no method of finding periodic motions (except relative
equilibria) of N vortices is known. We present one such method and find infinite
families of periodic motions on surfaces possessing certain symmetries, including
spheres, ellipsoids of revolution and cylinders. Our families exhibit bifurcations. N
can be made arbitrarily large. Numerical plots of bifurcations are given.