This research project seeks to explore specific groups of symmetries with metrizable universal minimal flows and attempt to determine whether all such groups are uniquely ergodic. Flows are continuous actions of a group of symmetries on compact Hausdorff topological spaces. For a group of symmetries to be uniquely ergodic means that there is a unique invariant measure on the space that applies to every minimal flow of the group of symmetries. There have been developments recently in the field of ergodic theory surrounding examples of these types of groups of symmetries, and how ones with metrizable universal minimal flows are uniquely ergodic, however, the field still lacks a generalization of this idea to all automorphism groups with such properties. This problem will first be approached by a survey of known results and an attempt to find combinatorial alternatives to the probabilistic methods originally used. Then a generalization of the ergodic result will be sought after to show all symmetry groups with the flow property are ergodic.