  These lectures are designed to provide a general introduction to the
Einstein-Vlasov system and to the global Cauchy problem for these equations. To
start with some general facts are collected and a local existence theorem for
the Cauchy problem stated. Next the case of spherically symmetric
asymptotically flat solutions is examined in detail. The approach taken, using
maximal-isotropic coordinates, is new. It is shown that if a singularity occurs
in the time evolution of spherically symmetric initial data, the first
singularity (as measured by a maximal time coordinate) occurs at the centre.
Then it is shown that for small initial data the solution exists globally in
time and is geodesically complete. Finally, the proof of the general local
existence theorem is sketched. This is intended to be an informal introduction
to some of the ideas which are important in proving such theorems rather than a
formal proof.
