The study of volumes of hyperbolic 3-manifolds has developed into a rich subject in which many different techniques interact. In these talks I will describe ongoing work with Marc Culler in which we bring some of the most classical techniques in 3-manifold topology, in combination with such recent developments as the proof of Marden’s conjecture and Perelman’s estimates for the Ricci flow with surgery, to bear on the investigation of connections between the volume of a hyperbolic 3-manifold and such classical invariants as homology and Heegaard genus.