The random walks in $R^d$ play the central role in analysis of many practically important stochastic systems (Cramer – Lundberg risk models in the insurance business, contact processes in the population dynamics, phase transition for the polymers etc.). Their distributions in the area of the “typical deviations” are the subject of the classical theory of
summation of i.i.d.r.v. But as a rule, in the applications we need the large deviation results. The corresponding theory for the light tails essentially goes to Cramer, but the large deviations for the heavy tailed random walks attracted the attention of the specialists just recently (the school of A. A. Borovkov and many others). The theory is not yet
complete, especially in multi-dimensional case. The part of the talk will contain a brief review of some recent results in this area. In the second part we will discuss the applications. They will include the analysis of the intermittency phenomena in the population dynamics and Cramer – Lundberg type theory based on the empirical data.