Abstract: We introduce and study the notion of essential dimension for linear algebraic groups defined over an algebraically closed fields of characteristic zero. The essential dimension is a numerical invariant of the group; it is often equal to the minimal number of independent parameters required to describe all algebraic objects of a certain type. For example, if our group G is S_n, these objects are field extensions, if G = O_n, they are quadratic forms, if G = PGL_n, they are division algebras (all of degree n), if G = G_2, they are octonion algebras, if G = F_4, they are exceptional Jordan algebras. We develop a general theory, then compute or estimate the essential dimension for a number of specific groups, including all of the above-mentioned examples. In the last section we give an exposition of results, communicated to us by J.-P. Serre, which relate the essential dimension of G to cohomological invariants of principal G-bundles.