The problem of singularities and chaos in cosmology

We consider different aspects of the problem of cosmological singularity such as the BKL oscillatory approach to the singularity, the new features of the cosmological dynamics in the neighbourhood of the singularity in multidimensional and superstring cosmological models and their connections with such a modern branch of mathematics as infinite-dimensional Lie algebras. The chaoticity of the oscillatory approach to the cosmological singularity is also discussed. The Conclusion contains some thoughts about the past and the future of the Universe in the light of the oscillatory approach to the Big Bang and the Big Crunch cosmological singularities.