Robot Programming by Demonstration (RbD), also referred to as Learning by Imitation, explores user-friendly means of teaching a robot new skills. Recent advances in RbD have identified a number of key-issues for ensuring a generic approach to the transfer of skills across various agents and contexts. This thesis focuses on the two generic questions of what-to-imitate and how-to-imitate, which are respectively concerned with the problem of extracting the essential features of a task and determining a way to reproduce these essential features in different situations. The perspective adopted in this work is that a skill can be described efficiently at a trajectory level and that the robot may infer what are the important characteristics of the skill by observing multiple demonstrations of it, assuming that the important characteristics are invariant across the demonstrations. The proposed approach is based on the use of well-established statistical methods in a RbD application, by using Hidden Markov Model (HMM) as a first approach and then moving on to the joint use of Gaussian Mixture Model (GMM) and Gaussian Mixture Regression (GMR). Even if these methods were applied extensively in various fields of research including human motion analysis and robotics, their use essentially focused on gesture recognition rather than on gesture reproduction. Moreover, the models were usually trained offline using large datasets. Thus, the use and combination of these machine learning techniques in a Learning by Imitation framework is challenging and has not been extensively studied yet. In this thesis, we show that these methods are well suited for incremental and active teaching scenarios where the user only provides a small number of demonstrations, either by wearing a set of motions sensors attached to his/her body, or by helping the robot refine its skill by kinesthetic teaching, that is, by embodying the robot and putting it through the motion. These techniques are then applied for enabling a Fujitsu HOAP-2 and HOAP-3 humanoid robot to extract automatically different constraints by observing several manipulation skills and to reproduce these skills in various situations through Gaussian Mixture Regression (GMR). The contributions of this thesis are threefold: (1) it contributes to RbD by proposing a generic probabilistic framework to deal with recognition, generalization, reproduction and evaluation issues, and more specifically to deal with the automatic extraction of task constraints and with the determination of a controller satisfying several constraints simultaneously to reproduce the skill in a different context; (2) it contributes to Human-Robot Interaction (HRI) by proposing active teaching methods that puts the human teacher "in the loop" of the robot's learning by using an incremental scaffolding process and by using different modalities to produce the demonstrations; (3) it finally contributes to robotics and HRI through various real-world applications of the framework showing learning and reproduction of communicative gestures and manipulation skills. The generality of the proposed framework is also demonstrated through its joint use with other learning approaches in collaborative experiments.