Let K(n) and E_n be the Morava K-theory and Morava E-theory of height n at prime p respectively. The famous result of Devinatz and Hopkins show that the homotopy groups of the K(n) localization of the sphere spectrum may be computed by a homotopy fixed point spectral sequence related to E_n. For a (not necessarily finite) spectra X, one might expect that the K(n) localization of X can be computed from a similar homotopy fixed point on E_n \wedge X. However, there is an obstruction due to the fact that the localization by K(n) is not a smashing localization for n > 0. Fortunately, a result of Hovey and Strickland shows that the K(n) localization of X can be, to an extent, recovered by a sequence of critical finite complexes called ``generalized Moorse spectra". This is the result we will survey in this talk.