  We introduce the spin Hecke algebra, which is a q-deformation of the spin
symmetric group algebra, and its affine generalization. We establish an algebra
isomorphism which relates our spin (affine) Hecke algebras to the (affine)
Hecke-Clifford algebras of Olshanski and Jones-Nazarov. Relation between the
spin (affine) Hecke algebra and a nonstandard presentation of the usual
(affine) Hecke algebra is displayed, and the notion of covering (affine) Hecke
algebra is introduced to provide a link between these algebras. Various
algebraic structures for the spin (affine) Hecke algebra are established.
