This is a joint Geometry/Topology day.  The past fifteen years have seen a great deal of progress towards a complete picture of hyperbolic manifolds of low volume. The volume of a hyperbolic manifold is a topological invariant and can be viewed as a measure of complexity. In fact, there are only finitely many hyperbolic manifolds of a given volume. For hyperbolic 3-manifolds with cusps, one can also consider the volume of the maximal horoball neighborhood of a cusp. In this talk, we will present preliminary results and techniques for understanding the infinite families of hyperbolic 3-manifolds of low cusp volume. These families are of particular interest as they exhibit the largest number of exceptional Dehn fillings. As in some other results on hyperbolic manifolds of low volume, our technique utilizes a rigorous computer assisted search. This talk will focus on providing sufficient background to explain the setup and goals of our approach. This work is joint with David Gabai, Robert Meyerhoff, Nathaniel Thurston, and Robert Haraway.