As part of his revolutionary work on hyperbolic geometry in
the 1970's, Thurston generalizing work of Jorgensen and Gromov, showed that the set of volumes of complete finite volume hyperbolic 3-manifolds is closed and well ordered. Recently, Robert Meyerhoff and Peter Milley and the speaker showed that the Weeks manifold is the unique lowest volume
closed orientable one, culminating a 30+ year effort by many mathematicians using a wide variety of geometric and topological techniques. This lecture will survey these and other developments and discuss some of the many open problems in the area.