In Haynes Miller's proof of the Sullivan conjecture on maps from classifying spaces, Quillen's derived functor notion of homology (in the case of commutative algebras) is a critical ingredient. This suggests that homology for the larger class of algebraic structures parametrized by an operad O will also provide interesting and useful invariants. Working in the context of symmetric spectra (resp. unbounded chain complexes), we introduce a (homotopy) completion tower for algebras over operads in symmetric spectra (resp. unbounded chain complexes). We prove that a weak equivalence on topological Quillen homology induces a weak equivalence on homotopy completion, and that for 0-connected algebras and modules over a -1-connected operad, the homotopy completion tower interpolates between topological Quillen homology and the identity functor. As a consequence we obtain a Whitehead Theorem for topological Quillen homology. This is part of a larger goal to attack the problem: how much of an O-algebra can be recovered from its topological Quillen homology? This talk is an introduction to these results (joint with K. Hess) with an emphasis on several of the motivating ideas.