**Title:** Effective hyperbolic geometry

**Abstract:** Powerful theorems of Thurston, Perelman, and Mostow tell us that almost every 3-dimensional manifold admits a hyperbolic metric, and that this metric is unique. Thus, in principle, there is a 1-to-1 correspondence between a combinatorial description of a 3-manifold and its geometry. The existence of this 1-to-1 correspondence has been known, at least conjecturally, for over 30 years. On the other hand, only in the last few years have we begun to see the outlines of a concrete dictionary between combinatorial features and geometric measurements. I will survey some of what is known and unknown, paying special attention to the problem of estimating the volume of a knot complement in the 3-sphere.