The emergence of various underwater wireless communication systems has been on the rise due to increasing human activities in the marine environment. In underwater wireless communication networks (UWCNs), several communication devices, such as sensors and autonomous underwater vehicles (AUV) are interconnected to expand communication coverage, monitoring, information gathering, and surveillance. These devices operate on batteries, making their replacement and recharging difficult. Consequently, sustaining the operational lifetime of UWCNs is deemed a major challenge. This leads to the development of various energy harvesting (EH) techniques to perpetuate the power supply to underwater devices. In this paper, we present a review of various energy sources and EH techniques applicable to UWCNs. To achieve this, we classify the energy sources into various categories in order to establish the peculiarities of each source and the type of harvester applicable to each category. Based on this classification, we present discussions on various contributions of articles related to applications of EH techniques in UWCNs. In addition to various insights gained from the presented papers, we establish that energy harvesters based on triboelectric effect, piezoelectric effect, sediment microbial fuel cell, acoustic, and optical power transfer are suitable for low-power (milliwatt-order) consuming devices such as sensors. Also, for devices with high power consumption requirement, such as AUV, solar and inductive power transfer-based harvesters should be employed. Furthermore, we identify several technical challenges that should be taken into consideration during the planning and system design phases. Finally, we highlight open research areas that could further improve the EH and communication processes in UWCNs.