Motivated by geometric realizations of (degenerate) affine Hecke algebras via convolution products on the equivariant K-theory (or homology) of the Steinberg variety, we define a "formal (affine) Hecke algebra" associated to any formal group law. Formal group laws are associated to algebraic oriented cohomology theories. When specialized to the formal group laws corresponding to K-theory and (co)homology, our definition recovers the usual affine and degenerate affine Hecke algebras. However, other formal group laws (such as those corresponding to elliptic and cobordism cohomology theories) give rise to apparently new algebras with interesting properties. This is joint work with Alex Hoffnung, Jose Malagon-Lopez, and Kirill Zainoulline.