Characteristic classes in generalized cohomology theories?

Stiefel-Whitney classes exist for any real-oriented cohomology theory. This is a (multiplicative) cohomology theory E equipped with an isomorphism

$E^* ( \mathbb{R} P^{\infty} ) \cong E^*(pt) [[x]]$

The two most well known examples are ordinary cohomology (i.e. singular cohomology) with $\mathbb{Z}/2$-coefficients and also unoriented bordism theory MO (this is the Thom spectrum MO, not "MathOverflow"). Note that such an isomorphism might not exist. For example it doesn't exist for ordinary Z cohomology, nor does it exist for K-theory.

The choice of this generator x is the first Stiefel-Whitney class and the other classes can be constructed using the splitting principle. For such a theory you will have Thom isomorphisms for all real bundles (not necessarily oriented).

I believe this is all explained in Switzer's book "Algebraic Topology", but I'm not sure. The course I learned it from didn't have a text book. This is often standard material for a second semester of graduate level algebraic topology, at least it was at UC Berkeley.

A well-studied example of this are the "cannibalistic characteristic classes" of Adams and Bott. If I remember correctly, there is a nice account of this in Bott's "Lectures on K(X)"; they are also discussed in Adams' J(X) papers (in Topology in the mid 60s).

The cannibalistic classes $\theta_k(V)$ of a complex bundle $V\to X$ are defined by the formula $$\psi^k(u) = \theta_k(V)\cdot u$$ where $\psi^k$ is an Adams operation, and $u$ is the standard Thom class in $K(\mathrm{Th}(V))$ (K-theory of the thom space of $V$). Thus, the role of Steenrod operations is played by the Adams operations.

The cannibalistic classes can be used to detect when two bundles $V$ and $V'$ are not "stably fiber homotopy equivalent"; i.e., when the sphere bundles $S(V\oplus \mathbb{R}^m)$ and $S(V'\oplus \mathbb{R}^m)$ are not fiber homotopy equivalent for all $m$. (See either of the references for this.)

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.