Renewable energy (RE)-powered base stations (BSs) have been considered as an attractive solution to address the exponential increasing energy demand in cellular networks while decreasing carbon dioxide (CO2) emissions. For the regions where reliable power grids are insufficient and infeasible to deploy, such as aerial platforms and harsh environments, RE has been an alternative power source for BSs. In this survey paper, we provide an overview of RE-enabled cellular networks, detailing their analysis, classification, and related works. First, we introduce the key components of RE-powered BSs along with their frequently adopted models. Second, we analyze the proposed strategies and design issues for RE-powered BSs that can be incorporated into cellular networks and categorize them into several groups to provide a good grasp. Third, we introduce feasibility studies on RE-powered BSs based on the recent literature. Fourth, we investigate RE-powered network components other than terrestrial BSs to address potential issues regarding RE-enabled networks. Finally, we suggest future research directions and conclusions.