Increasingly emerging technologies in microelectromechanical systems and wireless communications allows a mobile wireless sensor networks (MWSN) to be a more and more powerful mean in many applications such as habitat and environmental monitoring, traffic observing, battlefield surveillance, smart homes and smart cities. Nevertheless, due to sensor battery constraints, energy-efficiently operating a MWSN is paramount importance in those applications; and a plethora of approaches have been proposed to elongate the network longevity at most possible. Therefore, this paper provides a comprehensive review on the developed methods that exploit mobility of sensor nodes and/or sink(s) to effectively maximize the lifetime of a MWSN. The survey systematically classifies the algorithms into categories where the MWSN is equipped with mobile sensor nodes, one mobile sink or multiple mobile sinks. How to drive the mobile sink(s) for energy efficiency in the network is also fully reviewed and reported.