Energy efficiency and renewable energy are the main pillars of sustainability and environmental compatibility. This study presents an overview of sustainable and green cellular base stations (BSs), which account for most of the energy consumed in cellular networks. We review the architecture of the BS and the power consumption model, and then summarize the trends in green cellular network research over the past decade. As its major contribution, this study highlights the uses of renewable energy in cellular communication by: (i) investigating the system model and the potential of renewable energy solutions for cellular BSs; (ii) identifying the potential geographical locations for renewable-energy-powered BSs; (iii) performing case studies on renewable-energy-powered cellular BSs and suggesting future research directions based on our findings; (iv) examining the present deployment of sustainable and green BSs; and (v) studying the barriers that prevent the widespread use of renewable-energy-powered BSs and providing recommendations for future work.