  We give a general review of recent developments in the theory of vortices in
superfluids and superconductors, discussing why the dynamics of vortices is
important, and why some key results are still controversial. We discuss work
that we have done on the dynamics of quantized vortices in a superfluid.
Despite the fact that this problem has been recognized as important for forty
years, there is still a lot of controversy about the forces on and masses of
quantized vortices. We think that one can get unambiguous answers by
considering a broken symmetry state that consists of one vortex in an infinite
ideal system. We argue for a Magnus force that is proportional to the
superfluid density, and we find that the effective mass density of a vortex in
a neutral superfluid is divergent at low frequencies. We have generalized some
of the results for a neutral superfluid to a charged system.
