Cuspidal robots can move from one inverse or direct kinematic solution to another without ever passing through a singularity. These robots have remained unknown because almost all industrial robots do not have this feature. However, in fact, industrial robots are the exceptions. Some robots appeared recently in the industrial market can be shown to be cuspidal, but, surprisingly, almost nobody knows it and robot users meet difficulties in planning trajectories with these robots. This article proposes a review on the fundamental and application aspects of cuspidal robots. It addresses the important issues raised by these robots for the design and planning of trajectories. The identification of all cuspidal robots is still an open issue. This article recalls in details the case of serial robots with three joints, but it also addresses robots with more complex architectures such as 6-revolute-jointed robot and parallel robots. We hope that this article will help disseminate more wide knowledge on cuspidal robots.