  Theories, simulations and experiments on vortex dynamics in
quasi-two-dimensional magnetic materials are reviewed. These materials can be
modelled by the classical two-dimensional anisotropic Heisenberg model with XY
(easy-plane) symmetry. There are two types of vortices, characterized by their
polarization (a second topological charge in addition to the vorticity): Planar
vortices have Newtonian dynamics (even-order equations of motion) and exhibit
strong discreteness effects, while non-planar vortices have non-Newtonian
dynamics (odd-order equations of motion) and smooth trajectories. The influence
of thermal fluctuations on single vortices is investigated. Different types of
noise and damping are discussed and implemented into the microscopic equations
which yields stochastic equations of motion for the vortices. The solutions of
the these equations are compared with Langevin dynamics simulations. Moreover,
noise-induced transitions between opposite polarizations of a vortex are
investigated. For temperatures above the Kosterlitz-Thouless vortex-antivortex
unbinding transition, a phenomenological theory, namely the vortex gas
approach, yields central peaks in the dynamic form factors for the spin
correlations. Such peaks are observed both in combined Monte Carlo- and Spin
Dynamics-Simulations and in inelastic neutron scattering experiments. However,
the assumption of ballistic vortex motion appears questionable.
