Essential dimension of an algebraic object is the smallest number of algebraically independent parameters required to define the object. This notion was introduced by Buhler, Reichstein and Serre about 20 years ago.The relations to different parts of algebra such asalgebraic geometry, Galois cohomology and representation theory will be discussed.

 Oriented equivariant cohomology theories and the associated formal groups laws have been a subject of intensive investigations since 60's, mostly inspired by the theory of complex cobordism in topology. In the present talk we discuss several recent developments in the study of algebraic analogues of such theories, e.g. algebraic cobordism of Levine-Morel or algebraic elliptic cohomology, of projective homogeneous varieties. In particular, we address the problem of constructing the Schubert and the Bott-Samelson classes for such theories.