The increasing demand for spectral and energy efficient communication networks has spurred a great interest in energy harvesting (EH) cognitive radio networks (CRNs). Such a revolutionary technology represents a paradigm shift in the development of wireless networks, as it can simultaneously enable the efficient use of the available spectrum and the exploitation of radio frequency (RF) energy in order to reduce the reliance on traditional energy sources. This is mainly triggered by the recent advancements in microelectronics that puts forward RF energy harvesting as a plausible technique in the near future. On the other hand, it is suggested that the operation of a network relying on harvested energy needs to be redesigned to allow the network to reliably function in the long term. To this end, the aim of this survey paper is to provide a comprehensive overview of the recent development and the challenges regarding the operation of CRNs powered by RF energy. In addition, the potential open issues that might be considered for the future research are also discussed in this paper.