  We study standing wave solutions to nonlinear Schr{\"o}dinger equations, on a
manifold with a rotational symmetry, which transform in a natural fashion under
the group of rotations. We call these vortex solutions. They are higher
dimensional versions of vortex standing waves that have been studied on the
Euclidean plane. We focus on two types of vortex solutions, which we call
spherical vortices and axial vortices.
