Unmanned Aerial Vehicles have undergone rapid developments in recent decades. This has made them very popular for various military and civilian applications allowing us to reach places that were previously hard to reach in addition to saving time and lives. A highly desirable direction when developing unmanned aerial vehicles is towards achieving fully autonomous missions and performing their dedicated tasks with minimum human interaction. Thus, this paper provides a survey of some of the recent developments in the field of unmanned aerial vehicles related to safe autonomous navigation, which is a very critical component in the whole system. A great part of this paper focus on advanced methods capable of producing three-dimensional avoidance maneuvers and safe trajectories. Research challenges related to unmanned aerial vehicle development are also highlighted.