  Recent progress in understanding the structure of cosmological singularities
is reviewed. The well-known picture due to Belinskii, Khalatnikov and Lifschitz
(BKL) is summarized briefly and it is discussed what existing analytical and
numerical results have to tell us about the validity of this picture. If the
BKL description is correct then most cosmological singularities are
complicated. However there are some cases where it predicts simple
singularities. These cases should be particularly amenable to mathematical
investigation and the results in this direction which have been achieved so far
are described.
