In this work we introduce a type of hybrid equation closely related to those that appear when a digital controller or a hybrid controller of a certain class is interconnected with a full non-linear continuous-time system. For these equations we obtain results concerning existence, uniqueness and noncontinuability of solutions. This results are used to study the behaviour of the hybrid dynamical systems associated with those equations. We establish relationships between the stability of these hybrid dynamical systems and the ones corresponding to their associated discretized systems on one hand, and to their associated linearized systems on the other. We also characterize their stability, uniform asymptotic stability and exponential stability. These results are applied to the exponential stabilization of a non-linear control system by digital or hybrid controllers.