(Joint work with Robert Haraway, Robert Meyerhoff, Nathaniel Thurston and Andrew Yarmola) With rigorous computer assistance, both discrete and continuous, we show that if N is a complete finite volume hyperbolic 3-manifold with a maximal cusp of volume at most 2.62 then it is obtained by filling one of 16 explicit 2 or 3-cusped hyperbolic 3-manifolds. As an application, with more rigorous computer assistance, we (with Tom Crawford) show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non hyperbolic fillings.