The Universe evolves hierarchically with small structures merging and falling in to form bigger structures. Due to its proximity, the Local Group (LG) is the best place to witness and study these hierarchical processes in action as evidenced by e.g., the many stellar streams found around the Milky Way (MW) and M31. Stellar systems in the LG have therefore become the benchmark for testing many aspects of cosmological theories. Despite the advances in both observational and theoretical areas in the last decade or so, many fundamental properties (e.g., mass profiles) of the LG and its constituents still remain poorly understood. This is mainly due to the limited information on the transverse motions of stellar systems in the LG. Proper motion (PM) measurements are required to resolve this issue, but this has been technically challenging since the typical motions across the sky are very small. In the past few years, our team has developed a state-of-the-art technique to determine proper motions of resolved stellar systems using multi-epoch HST data. In this talk, I will present results from our HST projects for measuring proper motions of stellar systems in the LG, and discuss how these results are improving our understanding of the LG dynamics.