The Friedman-Lamaitre-Robertson-Walker metric of a homogeneous and isotropic universe is generally agreed upon today as the cosmological model. However, the question of why the observable universe is so homogeneou and isotropic still remains an open problem. To answer  it, one must try to analyze the possible behaviour of space-time near a cosmological singularity. One of the attempts at such studies was the Balinskii-Khalatnikov-Lifshitz conjecture, which postulates that near the singularity time derivatives of the metric dominate heavily over spatial derivaives and matter fields. This in turn suggests consideration of spatially homogeneous, but anisotropic space-time models as good approximations.

I will begin my talk by discussing the general structure of such space-times and showing that their properties are almost entirely determined by the Lie algebra of their spatial Killing fields. This allows them to be classified by the Bianch classification of the three-dimensional Lie algebras. Several examples of particular physical interest will be examined in detail - although still relatively simple, some of these solutions of the Einstein equations display very untrivial properties, such as modelling an isotropisation process or even undergoing chaotic oscillations when approaching the singularity.