  We review the intimate connection between (super-)gravity close to a
spacelike singularity (the "BKL-limit") and the theory of Lorentzian Kac-Moody
algebras. We show that in this limit the gravitational theory can be
reformulated in terms of billiard motion in a region of hyperbolic space,
revealing that the dynamics is completely determined by a (possibly infinite)
sequence of reflections, which are elements of a Lorentzian Coxeter group. Such
Coxeter groups are the Weyl groups of infinite-dimensional Kac-Moody algebras,
suggesting that these algebras yield symmetries of gravitational theories. Our
presentation is aimed to be a self-contained and comprehensive treatment of the
subject, with all the relevant mathematical background material introduced and
explained in detail. We also review attempts at making the infinite-dimensional
symmetries manifest, through the construction of a geodesic sigma model based
on a Lorentzian Kac-Moody algebra. An explicit example is provided for the case
of the hyperbolic algebra E10, which is conjectured to be an underlying
symmetry of M-theory. Illustrations of this conjecture are also discussed in
the context of cosmological solutions to eleven-dimensional supergravity.
