  In this paper we propose a systematic study of Thom polynomials for group
actions defined by M. Kazarian. On one hand we show that Thom polynomials are
first obstructions for the existence of a section and are connected to several
problems of topology, global geometry and enumerative algebraic geometry. On
the other hand we describe a way to calculate Thom polynomials: the method of
restriction equations. It turned out that though the idea is quite simple the
method is very powerful. We reproduced and improved earlier result in several
directions (singularities, Schubert calculus, quivers). However a proper
introduction to the basic theorems was missing. In this paper we try to pay
this debt as well as we present the connections with obstruction theory and
equivariant cohomology. We give some new results and outline possible
generalizations and problems.
