TITLE: Characteristic classes in generalized cohomology theories?
QUESTION [17 upvotes]: Hello,
'ordinary' Stiefel-Whitney classes are elements of the singular cohomology ring and are constructed using the Thom isomorphism and Steenrod squares. So I think they should exist for any (generalized) multiplicative cohomology theory for which  the Thom isomorphism and cohomology operations like the Steenrod squares exist. If I am not wrong, I would be really happy about some references on this.
Thanks in advance (and merry christmas)
Jonas

REPLY [6 votes]: Maybe this isn't the "right" way to think about them, but I have often found the following characterization much more appealing: the characteristic classes are just elements of $E^*(BG)$ for some cohomology theory and some group $G$, if $E$ is ordinary cohomology with integer coefficients and $G$ is the infinite unitary group you get Chern classes etc. The universal property of $BG$ gives you a unique map that classifies each $G$-bundle, now look at the map in cohomology that you get from this and you get the characteristic class of the desired bundle. This probably requires some sort of orientability for things to be "nice" but you can certainly define characteristic classes for any cohomology theory this way, they just might not have nice relations.