The seamless integration of low-power, miniaturised, invasive/non-invasive lightweight sensor nodes have contributed to the development of a proactive and unobtrusive Wireless Body Area Network (WBAN). A WBAN provides long-term health monitoring of a patient without any constraint on his/her normal dailylife activities. This monitoring requires the low-power operation of invasive/non-invasive sensor nodes. In other words, a power-efficient Medium Access Control (MAC) protocol is required to satisfy the stringent WBAN requirements, including low-power consumption. In this paper, we first outline the WBAN requirements that are important for the design of a low-power MAC protocol. Then we study low-power MAC protocols proposed/investigated for a WBAN with emphasis on their strengths and weaknesses. We also review different power-efficient mechanisms for a WBAN. In addition, useful suggestions are given to help the MAC designers to develop a low-power MAC protocol that will satisfy the stringent requirements.