The oriented movement of biological cells or organisms in response to a chemical gradient
is called chemotaxis. The most interesting situation related to self-organization
phenomenon takes place when the cells detect and response to a chemical which is secreted
by themselves. Since pioneering works of Patlak (1953) and Keller and Segel (1970) many
particularized models have been proposed to describe the aggregation phase of this
process. Most of efforts were concentrated, so far, on mathematical models in which the
formation of aggregate is interpreted as finite time blow-up of cell density. In recently
proposed models cells are no more treated as point masses and their finite volume is
accounted for. Thus, arbitrary high cell densities are precluded in such description and a
threshold value for cells density is a priori assumed. Different modeling
approaches based on this assumption lead to a class of quasilinear parabolic systems with
strong nonlinearities including degenerate or singular diffusion. We give a survey of
analytical results on the existence and uniqueness of global-in-time solutions, their
convergence to stationary states and on a possibility of reaching the density threshold by
a solution. Unsolved problems are pointed as well.