Small cell networks (SCNs) have attracted great attention in recent years due to their potential to meet the exponential growth of mobile data traffic and the increasing demand for better quality of service and user experience in mobile applications. Nevertheless, a wide deployment of SCNs has not happened yet because of the complexity in the network planning and optimization, as well as the high expenditure involved in deployment and operation. In particular, it is difficult to provide grid power supply to all the small cell base stations (SCBSs) in a cost effective way. Moreover, a dense deployment of SCBSs, which is needed to meet the capacity and coverage of the next generation wireless networks, will increase operators' electricity bills and lead to significant carbon emission. Thus, it is crucial to exploit off-grid and green energy sources to power SCNs, for which energy harvesting (EH) technology is a viable solution. In this article, we will conduct a comprehensive study of EH-SCNs, and investigate important aspects, including the feasibility analysis, network deployment, and network operation issues. The advantages, as well as unique challenges, of EH-SCNs will be highlighted, together with potential solutions and effective design methodologies.