Abstract: Quillen’s derived functor notion of homology provides interesting and useful invariants in a wide variety of homotopical algebraic contexts. For instance, in Haynes Miller’s proof of the Sullivan conjecture on maps from classifying spaces, Quillen homology of commutative algebras (Andre-Quillen homology) is a critical ingredient. Working in the topological context of structured ring spectra, this talk will introduce several recent results on localization and completion with respect to topological Quillen homology of commutative ring spectra (topological Andre-Quillen homology), $E_n$ ring spectra, and operad algebras in spectra. This includes homotopical analysis of a completion construction and strong convergence of its associated homotopy spectral sequence. The localization and completion constructions for structured ring spectra are precisely analogous to Sullivan's localization and completion of spaces (for which he recently won the Wolf prize), and Bousfield-Kan's version of Sullivan's localization and completion called the $R$-completion of a space with respect to a ring $R$. This is joint work with Michael Ching.