TITLE: Does a "Chern character" exist for any generalized cohomology theory?
QUESTION [23 upvotes]: The Chern character is a ring homomorphism from complex K-theory to the usual cohomology.
1) I wonder if there are "Chern character"-like ring homomorphisms from other generalized cohomology theories to the usual cohomology. Are they related with Atiyah-Hirzebruch?
2) And if there are such nice homomorphisms, what is the "Todd genus" in these cases, making the generalization of that famous diagram in Grothendieck–Hirzebruch–Riemann–Roch commute?
When I think about it, I cannot even recall seeing anything like this in real K-theory, but that is probably because I don't really know real K-theory at all.

REPLY [30 votes]: In Oscar Randall-Williams' answer "connective" is unnecessary. Also, there is no need to choose that isomorphism (from a rational theory to the ordinary theory with the same coefficient groups); it is canonical.
The generalization of the Todd genus or Todd class arises when the multiplicative theory $E$ has a "complex orientation": a multiplicatively well-behaved way of producing Thom isomorphisms in $E^*$-theory for all complex vector bundles.