Determining whether a given finitely generated group of isometries is discrete is a formidable problem. Let $\Gamma$ be a rank 2 non-elementary subgroup of PSL(2,R); J. Gilman and B. Maskit developed a discreteness algorithm which codified previously existing algorithms for all such $\Gamma$. We intend to motivate the discreteness problem, give a synopsis of the Gilman-Maskit algorithm, and share some efforts toward developing discreteness algorithms for higher rank groups. In particular, a discreteness algorithm for $\Gamma$ (as above) except generated by 3 parabolic isometries will be presented. Speaker(s): Caleb Ashley (UM)