Banach algebras is a setting in which knowledge about the structure of
a Banach algebra is obtained by classifying its actions on Banach
spaces. For Banach subalgebras of B(X), the Banach algebra of bounded
operators on a Banach space X, this task roughly amounts to a study of
interdependence of factorization properties of linear operators and
geometric structure of the underlying Banach space. In the talk we will
describe the general ideas and give some recent developments.