Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is orientable, there is a right-half-plane spectral sequence $(E^r,d^r)$ with second page $$E^2_{p,q} = H_p\big(B;h_q(F)\big)$$ and converging to $h_*(E)$, first mentioned in Atiyah and Hirzebruch's seminal paper on complex K-theory but apparently folklore since work of G. W. Whitehead.
Likewise, given a generalized cohomology theory $h^*$, there is $(E_r,d_r)$ with $$E^{p,q}_2 = H^p\big(B;h^q(F)\big)$$ which converges to $h^*(E)$ if $B$ is a finite CW complex. The more careful descriptions I've read of this sequence seem to concur that in full generality, convergence does not hold. It seems there are (at least) two potential pitfalls:
The sequence may converge to something other than $h^*(E)$.
The sequence may fail to converge at all.
Under what conditions does the cohomological sequence converge to $h^*(E)$?