Abstract This review concentrates on three main challenges of controlling a quadrotor unmanned aerial vehicle (UAV) as an underactuated mechanical system (UMS). The challenges include underactuation, model uncertainty, and actuator failure. The review also illustrates the state of the quadrotor control research in an attempt to find practical solutions for those challenges. By definition, a UMS has less number of control actions than the number of degrees of freedom to be controlled. Hence, traditional control strategies developed for fully actuated systems are not directly applicable to UMS. Research on UMS is one of the most interesting topics in the robotics and control community because of its relevance to a variety of applications. Examples include humanoid robots and most aerial and underwater vehicles. However similar to any other nonlinear systems, the stability and control of quadrotors suffer from intrinsic complexities exasperated by underactuation, model uncertainties and actuator failure. In this review article, we are going to address these problems and discuss the solutions mostly concerning the application of different control strategies presented in recent literature. We also present the frontiers and research directions towards improving performance of quadrotors for emerging and complex applications.