Two different approaches to moduli theory in algebraic geometry have developed in parallel since the 1960's: The intrinsic approach attempts to describe a moduli problem as an algebraic stack, and to generalize the notions of algebraic geometry in order to study the geometry of this stack directly. The other approach is to approximate your moduli problem as the classification of orbits for the action of a reductive group on a projective variety (e.g., a Quot scheme), then turn the crank of geometric invariant theory (GIT) to identify a semistability condition and construct a moduli space, and finally to re-express the resulting semistability condition more intrinsically. Over the last decade, work of myself and others has unified these two approaches in a package of theorems that allow one to study semistability conditions, moduli spaces, and canonical stratifications all in the intrinsic context. This has shed new light on several moduli problems of interest in algebraic geometry, such as the moduli of Fano varieties in higher dimensions or the moduli of decorated principal bundles on curves. I will survey techniques that have been developed to apply the general machinery in several contexts, as well as contexts where new techniques are needed.