Autonomous mission planning for unmanned aerial vehicles (UAVs) aims to leverage the capabilities of UAVs equipped with on-board sensors to accomplish a wide range of applications, including planetary exploration where greater science yields can be achieved at lower costs over shorter time periods. A significant body of research has already been performed with the aim of improving the autonomy of UAV missions, particularly in the areas of navigation and target identification. In this work, we review current approaches to drone navigation and exploration for planetary missions, with a focus on Mars and the main autonomy levels/techniques employed to achieve these levels. Recognising the importance of astrobiology in Mars exploration, we highlight progress in the area of autonomous biosignature detection capabilities trialed on Earth, and discuss the objectives and challenges in relation to future missions to Mars. Finally, we indicate currently available software tools and future work to improve autonomous mission planning capabilities.