Robot PbD started about 30 years ago, growing importantly during the past decade. The rationale for moving from purely preprogrammed robots to very flexible user-based interfaces for training the robot to perform a task is three-fold. First and foremost, PbD, also referred to as {\em imitation learning} is a powerful mechanism for reducing the complexity of search spaces for learning. When observing either good or bad examples, one can reduce the search for a possible solution, by either starting the search from the observed good solution (local optima), or conversely, by eliminating from the search space what is known as a bad solution. Imitation learning is, thus, a powerful tool for enhancing and accelerating learning in both animals and artifacts. Second, imitation learning offers an implicit means of training a machine, such that explicit and tedious programming of a task by a human user can be minimized or eliminated (Figure \ref{fig:what-how}). Imitation learning is thus a ``natural'' means of interacting with a machine that would be accessible to lay people. And third, studying and modeling the coupling of perception and action, which is at the core of imitation learning, helps us to understand the mechanisms by which the self-organization of perception and action could arise during development. The reciprocal interaction of perception and action could explain how competence in motor control can be grounded in rich structure of perceptual variables, and vice versa, how the processes of perception can develop as means to create successful actions. PbD promises were thus multiple. On the one hand, one hoped that it would make the learning faster, in contrast to tedious reinforcement learning methods or trials-and-error learning. On the other hand, one expected that the methods, being user-friendly, would enhance the application of robots in human daily environments. Recent progresses in the field, which we review in this chapter, show that the field has make a leap forward the past decade toward these goals and that these promises may be fulfilled very soon.