Abstract: Path planning and trajectory planning are crucial issues in the field of Robotics and, more generally, in the field of Automation. Indeed, the trend for robots and automatic machines is to operate at increasingly high speed, in order to achieve shorter production times. The high operating speed may hinder the accuracy and repeatability of the robot motion, since extreme performances are required from the actuators and the control system. Therefore, particular care should be put in generating a trajectory that could be executed at high speed, but at the same time harmless for the robot, in terms of avoiding excessive accelerations of the actuators and vibrations of the mechanical structure. Such a trajectory is defined as smooth. For such reasons, path planning and trajectory planning algorithms assume an increasing significance in robotics. Path planning algorithms generate a geometric path, from an initial to a final point, passing through pre-defined via-points, either in the joint space or in the operating space of the robot, while trajectory planning algorithms take a given geometric path and endow it with the time information. Trajectory planning algorithms are crucial in Robotics, because defining the times of passage at the via-points influences not only the kinematic properties of the motion, but also the dynamic ones.Namely, the inertial forces (and torques), to which the robot is subjected, depend on the accelerations along the trajectory, while the vibrations of its mechanical structure are basically determined by the values of the jerk (i.e. the derivative of the acceleration). Path planning algorithms are usually divided according to themethodologies used to generate the geometric path, namely: • roadmap techniques • cell decomposition algorithms • artificial potential methods. The algorithms for trajectory planning are usually named by the function that is optimized, namely: • minimum time • minimum energy • minimum jerk. Examples of hybrid algorithms, which optimize more than a single function, are also found in the scientific literature. In this chapter, the general problem of path planning and trajectory planning will be addressed, and an extended overview of the algorithms belonging to the categories mentioned above will be carried out, with references to the numerous contributions to this field.