Many nonlinear systems have a striking phenomenon, an explosive instability, which occurs if the system is linearly unstable and nonlinearity does not saturate an exponential growth of small perturbations, but, on the contrary, results in singularity formation in a finite time. Near singularity point there is usually a qualitative change in underlying nonlinear phenomena, reduced models loose their applicability and other mechanisms become important such as inelastic collisions in the Bose-Einstein condensate; optical breakdown and dissipation in nonlinear optical media and plasma; wave breaking in hydrodynamics. Special focus will be on singularity formation in Keller-Segel model which describes macroscopic behavior of bacterial colonies. Bacteria secret chemical which attract other bacteria so that they move along chemical gradient. Keller-Segel model is a parabolic-elliptic system of partial differential equations for bacterial density and chemical concentration. If bacterial density exceed critical value then the density blows up in a finite time which corresponds to bacterial aggregation. In two dimensions collapse is critical and qualitatively it shares many common properties with critical collapse of nonlinear Schroedinger equation. However there are also serious differences, in particular because Keller-Segel model is not the Hamiltonian system. The objective of this work is to find out how blow up occurs. The self-similar solutions near blow up point are studied together with time dependence of these solutions and their stability.