Abstract: In the early 2000s, Frigerio, Martelli, and Petronio studied 3-manifolds of smallest combinatorial complexity that admit hyperbolic structures. As part of this work they defined and studied the class \(M_{g ,k}\) of smallest complexity manifolds having \(k\) torus cusps and connected totally geodesic boundary a surface of genus \(g\) . In this talk, I will survey some of their important results on the construction of manifolds in \(M_{g ,k}\) and then present a complete classification of the manifolds in \(M_{k ,k}\) and \(M_{k+1 ,k}\). For a fixed \(k\), these are the cases when \(g\) is as small as possible. This is joint work with Max Forester and Nick Miller.