Complex systems typically possess a hierarchical structure, characterized by continuous-variable dynamics at the lowest level and logical decision-making at the highest. Virtually all control systems today perform computer-coded checks and issue logical as well as continuous-variable control commands. Such are 'hybrid' systems. Traditionally, the hybrid nature of these systems is suppressed by converting them into either purely discrete or continuous entities. Motivated by real-world problems, we introduce 'hybrid systems' as interacting collections of dynamical systems, evolving on continuous-variable state spaces, and subject to continuous controls and discrete phenomena. We identify the discrete phenomena that arise in hybrid systems and review previously proposed models. We propose a hybrid control model, coupling differential equations and automata, that encompasses them. Our unified model is natural for posing and solving hybrid analysis and control problems. We discuss topological issues that arise in hybrid systems analysis. Then we compare the computational capabilities of analog, digital, and hybrid machines by proposing intuitive notions of analog machines simulating digital ones. We show that simple continuous systems possess the power of universal computation. Hybrid systems have further simulation capabilities. For instance, we settle the famous asynchronous arbiter problem in both continuous and hybrid settings. Further, we develop analysis tools for limit cycle existence, perturbation robustness, and stability. We analyze a hybrid control system, typically used in aircraft, that logically switches between two conventional controllers. Stability of such systems has previously only been tested using extensive simulation; we prove global asymptotic stability for a realistic set of cases. Our tools demonstrate robustness of this stability with respect.