The character of a representation enumerates the eigenvalues of the operators by which a group G acts. A geometric generalization of a linear representation is an action of a group on a vector bundle V over some base X. Given an element g of G, we can ask how it acts on fibers over points fixed by g; we can also consider the action of g on global sections and, more generally, higher cohomology groups of V. The localization theorem of Atiyah and Bott is the first is a long series of important results connecting the two, and it will be the starting point of our discussion. My next goal will be to explain why is it interesting in enumerative geometry and which form does it take there.