The existence problem for periodic orbits is one of the central questions in
Hamiltonian dynamical systems. For systems describing the motion of a
charged particle in a magnetic field, i.e., for the so-called twisted
geodesic flows, this question was first addressed by V.I. Arnold in the
early 80s. Arnold's work initiated an extensive study of the existence
problem for periodic orbits of twisted geodesic flows via variational and
dynamical systems methods as well as in the context of symplectic topology.

In this talk we will discuss various aspects of this problem and related
results, focusing, in particular, on recent theorems obtained by symplectic
topological (or, more precisely, Floer homological) methods.