  We study a generalization of graded Hecke algebras introduced by Drinfeld in
1986, in which the role of a finite group G is played by a reductive algebraic
group. This includes continuous generalizations of symplectic reflection
algebras and of rational Cherednik algebras. We prove direct and converse PBW
theorems, and also study versions of our algebras in which the group G is
replaced with its Lie algebra. We also begin to develop the representation
theory of continuous Hecke algebras, which unifies representation theories of
real reductive groups, Drinfeld-Lusztig degenerate affine Hecke algebras, and
symplectic reflection (in particular, Cherednik) algebras. The paper is
dedicated to V. Drinfeld's 50-th birthday.
