  The structure of the general, inhomogeneous solution of (bosonic)
Einstein-matter systems in the vicinity of a cosmological singularity is
considered. We review the proof (based on ideas of
Belinskii-Khalatnikov-Lifshitz and technically simplified by the use of the
Arnowitt-Deser-Misner Hamiltonian formalism) that the asymptotic behaviour, as
one approaches the singularity, of the general solution is describable, at each
(generic) spatial point, as a billiard motion in an auxiliary Lorentzian space.
For certain Einstein-matter systems, notably for pure Einstein gravity in any
spacetime dimension D and for the particular Einstein-matter systems arising in
String theory, the billiard tables describing asymptotic cosmological behaviour
are found to be identical to the Weyl chambers of some Lorentzian Kac-Moody
algebras. In the case of the bosonic sector of supergravity in 11 dimensional
spacetime the underlying Lorentzian algebra is that of the hyperbolic Kac-Moody
group E(10), and there exists some evidence of a correspondence between the
general solution of the Einstein-three-form system and a null geodesic in the
infinite dimensional coset space E(10)/K (E(10)), where K (E(10)) is the
maximal compact subgroup of E(10).
