The groups $F \leq T \leq V$ were defined by Richard Thompson in 1965 and used to construct finitely presented groups with unsolvable word problems. $T$ and $V$ were also the first examples of infinite, finitely presented simple groups. Since then, these groups have been studied extensively using a rich interplay of algebraic, topological, and dynamical approaches. I will discuss recent work regarding the higher dimensional analogues of Thompson groups, $nV$, including the fact that $mV$ is not isomorphic to $nV$ for $n \neq m$, and that for every $n$ the group $nV$ is finitely presented and simple. The only background required for this talk is basic group theory.