  Spatially homogeneous cosmological models reduce to Hamiltonian systems in a
low dimensional Minkowskian space moving on the total energy shell $H=0$. Close
to the initial singularity some models (those of Bianchi type VIII and IX) can
be reduced further, in a certain approximation, to a non-compact triangular
billiard on a 2-dimensional space of constant negative curvature with a
separately conserved positive kinetic energy. This type of billiard has long
been known as a prototype chaotic dynamical system. These facts are reviewed
here together with some recent results on the energy level statistics of the
quantized billiard and with direct explicit semi-classical solutions of the
Hamiltonian cosmological model to which the billiard is an approximation. In
the case of Bianchi type IX models the latter solutions correspond to the
special boundary conditions of a `no-boundary state' as proposed by Hartle and
Hawking and of a `wormhole' state.
