The aerial manipulator is a special and new type of flying robot composed of a rotorcraft unmanned aerial vehicle (UAV) and a/several manipulator/s. It has gained a lot of attention since its initial appearance in 2010. This is mainly because it enables traditional UAVs to conduct versatile manipulating tasks from air, considerably enriching their applications. In this survey, a complete and systematic review of related research on this topic is conducted. First, various types of structure designs of aerial manipulators are listed out. Subsequently, the modeling and control methods are introduced in detail from the perspective of two types of typical application cases: free-flight and motion-restricted operations. Finally, challenges for future research are presented.