Connectivity and automation in vehicles provide the most intriguing opportunity for enabling users to better monitor transportation network conditions and make better operating decisions to improve safety and reduce pollution, energy consumption, and travel delays. This paper investigates the implications of optimally coordinating vehicles that are wirelessly connected to each other in roundabouts to achieve a smooth traffic flow. We apply an optimization framework and an analytical solution that allows optimal coordination of vehicles for merging in such traffic scenario. The effectiveness of the proposed approach is validated through simulation and it is shown that fully coordinated vehicles reduce total travel time by 51% and fuel consumption by 35%. Furthermore, we study the influence of vehicle coordination in a mixed-traffic environment and compare the network performance under different market penetration rates of connected and automated vehicles (CAVs). For this particular study with near-capacity demand, due to extremely unstable traffic, the results show that even with high penetration of CAVs (e.g., 80%), travel time and fuel savings are much less than a network of CAVs.