The task of modeling and controlling the evolution of dynamical sys- tems is one of the main objectives in mechatronics engineering. When approaching the problem of controlling physical or digital systems, the dynamical models have been historically divided into continuous-time, described by differential equations, and discrete-time, described by difference equations. In the last decade, a new class of models, known as hybrid dynamical systems, has gained popularity in the control community because of its high versatility. This framework combines continuous-time and discrete- time evolution, thus allowing for both the description of a broader class of systems and the achievement of better-performing controllers, compared to the traditional continuous-time alternatives. After the first rigorous introduction of the framework, several Lyapunov-based results were published in the literature, and numerous application areas were shown to benefit from the introduction of a hybrid dynamics, like systems involving impacts or physical systems connected to digital controllers (cyber-physical systems). In this thesis, we use the hybrid framework to study different mechatronics-inspired control problems. The applications we consider are diverse, so we split the presentation into three parts. In the first part we further analyze a particular hybrid control strategy, known as reset control, providing some new theoretical guarantees, together with an application to adaptive control. In the second part we consider two applications of the hybrid framework to the network dynamics field, specifically we analyze the problems of distributed state estimation and of uniform synchronization of nonlinear oscillators. In the third part, we use a hybrid approach to study two applications where this framework has been rarely employed, or not at all, namely smart agriculture and trajectory tracking for a bipedal walking robot. We study these application-inspired problems from a theoretical point of view, giving robust Lyapunov-based stability guarantees. We complement the theoretical analysis with numerical results, obtained from simulations or from experiments.