This chapter gives an informal introduction to hybrid dynamical systems and illustrates by simple examples the main phenomena that are encountered due to the interaction of continuous and discrete dynamics. References to numerous applications show the practical importance of hybrid systems theory. What is a hybrid system?Wherever continuous and discrete dynamics interact, hybrid systems arise. This is especially profound in many technological systems, in which logic decision making and embedded control actions are combined with physical processes. To capture the evolution of these systems, mathematical models are needed that combine in one way or another the dynamics of the continuous parts of the system with the dynamics of the logic and discrete parts. These mathematical models come in all kinds of variations, but basically consist of some form of differential or difference equations on the one hand and automata or other discrete-event models on the other hand. The collection of analysis and synthesis techniques based on these models forms the research area of hybrid systems theory, which plays an important role in the multi-disciplinary design of many technological systems that surround us. Three reasons to study hybrid systemsThe reasons to study hybrid systems can be quite diverse. Here we will provide three sources of motivation, which are related to (i) design of technological 2 Introduction to hybrid systems systems, (ii) networked control systems, and (iii) physical processes exhibiting non-smooth behavior.Challenges of multi-disciplinary design. When designing a technological system (Fig. 1.1) such as a wafer stepper, electron microscope, copier, robotic system, fast component mounter, medical system, etc., multiple disciplines need to make the overall design in close co-operation. For instance, the electronic design, mechanical design, and software design together have to result in a consistent, functioning machine. The designs are typically made in parallel by multiple groups of people, where the communication between these groups is often hampered by lack of common understanding and common models. The lack of common models complicates the making of cross-disciplinary design decisions that may have advantages for one discipline, but disadvantages for others. To make a good trade-off, the overall effect of such a design decision has to be evaluated as early as possible. As the complexity of a technological system with typically millions of lines of code and tens of thousands of mechanical components gives rise to many cross-disciplinary design decisions, a framework is required that supports efficient evaluation of design decisions incorporating quantitative information and models from multiple disciplines. Hybrid systems theory studies the behavior of dynamical systems, including the above mentioned technological systems, the modeling formalisms that involve both continuous models such as differential or difference equations describing the physical and mechanical part, and discrete models such as fin...