In this dissertation, we study the stability of hybrid composite dynamical systems. Such systems are composite systems consisting of a plant which is described by an input-output representation and a controller which is described by a state space representation;The plant is described by an operator (usually a differential equation or a partial differential equation). The controller may be a continuous time system or a discrete time system. In the continuous time case the controller is described by a set of ordinary differential equations, while in the discrete time case the controller is described by a set of difference equations. The latter case is of great importance since modern control systems are often computer controlled systems;We establish results for the well-posedness, attractivity, asymptotic stability in the large and exponential stability in the large for both continuous and discrete time cases. The applicability of our results is demonstrated by examples.