The rapidly enhancing sensing capabilities of smartphones are enabling the development of a wide range of innovative mobile sensing applications that are impacting on everyday life of mobile users. However, supporting long-term sensing applications is challenging because of their key requirements for continuous access to embedded sensors for gathering raw data, which can deplete the device’s battery in a few hours. This problem is expected to remain in the near future because the improvements on the capacity of batteries are coming at a slower pace than those advances in computing and sensing capabilities. The research community has highlighted the need for power-aware and context-aware sensing techniques deployed at different levels of mobile platforms for making a more efficient use of energy resources. Previous studies have analyzed the optimization of power consumption in mobile devices over different critical axes, like data transmission, computing, and hardware design. However, a comprehensive study focused in the challenges of power-aware smartphone-based sensing and strategies for addressing them has not been produced yet. This survey aims to fill this void with a particular focus on mobility sensing systems (e.g., human activity recognition, location-based services), presenting a comprehensive review of relevant strategies aimed at solving this issue. Also, this survey defines a taxonomy for such solutions, highlighting their strengths and limitations. Finally, most relevant open challenges and trends are discussed for providing insights for future research in the field.