  In this paper we prove the existence of vortices, namely standing waves with
non null angular momentum, for the nonlinear Klein-Gordon equation in dimension
$N\geq 3$. We show with variational methods that the existence of these kind of
solutions, that we have called \emph{hylomorphic vortices}, depends on a
suitable energy-charge ratio. Our variational approach turns out to be useful
for numerical investigations as well. In particular, some results in dimension
N=2 are reported, namely exemplificative vortex profiles by varying charge and
angular momentum, together with relevant trends for vortex frequency and
energy-charge ratio. The stability problem for hylomorphic vortices is also
addressed. In the absence of conclusive analytical results, vortex evolution is
numerically investigated: the obtained results suggest that, contrarily to
solitons with null angular momentum, vortex are unstable.
