Stability of motion is a central theme in the dynamics of mechanical systems. While the stability theory for systems with bilateral constraints is a well-established field, this monograph represents a systematic study of mechanical systems with unilateral constraints, such as unilateral contact, impact and friction. Such unilateral constraints give rise to non-smooth dynamical models for which stability theory is developed in this work. The book starts with the treatise of the mathematical background on non-smooth analysis, measure and integration theory and an introduction to the field of non-smooth dynamical systems. The unilateral constraints are modelled in the framework of set-valued force laws developed in the field of non-smooth mechanics. The embedding of these constitutive models in the dynamics of mechanical systems gives rises to dynamical models with impulsive phenomena. This book uses the mathematical framework of measure differential inclusions to formalise such models. The book proceeds with the presentation of stability results for measure differential inclusions. These stability results are then applied to nonlinear mechanical systems with unilateral constraints. The book closes with the study of the convergence property for a class of measure differential inclusions; a stability property for systems with time-varying inputs which is shown to be highly instrumental in the context of the control of mechanical systems with unilateral constraints. While the book presents a profound stability theory for mechanical systems with unilateral constraints, it also has a tutorial value on the modelling of such systems in the framework of measure differential inclusions. The work will be of interest to engineers, scientists and students working in the field of non-smooth mechanics and dynamics.