By virtue of distinguished wing shape morphing characteristics, the unrivaled agility and flight maneuverability of bats have inspired scientists and engineers to develop novel forms of robots that can fly like bats. The unique wing conformations, flight kinematics, and aerodynamics offer significant advantages over the conventional form of miniature air vehicle in terms of quiet, safe operations, improved efficiency, and enhanced maneuverability. Meanwhile, they also pose substantial challenges for robot design from multiple perspectives, including mechanical design, sensing, control, etc. The practical benefits and technical bottleneck have motivated the development of bat-inspired robots in recent years. The purpose of this paper is to summarize the designing principles and report current state-of-the-art of bat-inspired robot designs, emphasizing the respective distinguishing features of each paradigm, along with the room for further improvement. Rather than showcasing advancement in wing materials, we will focus on the mechanical design and control methodology. This paper will help researchers new in this realm to get familiar with the bat-inspired robots by adopting features from existing designs. It also concludes technical challenges associated with future development, involving biological research, aerodynamic modeling, mechanical design, and control technique.