This monograph presents the Timed Input/Output Automaton (TIOA) modeling framework, a basic mathematical framework to support description and analysis of timed (computing) systems. Timed systems are systems in which desirable correctness or performance properties of the system depend on the timing of events, not just on the order of their occurrence. Timed systems are employed in a wide range of domains including communications, embedded systems, real-time operating systems, and automated control. Many applications involving timed systems have strong safety, reliability, and predictability requirements, which make it important to have methods for systematic design of systems and rigorous analysis of timing-dependent behavior. The TIOA framework also supports description and analysis of timed distributed algorithms -- distributed algorithms whose correctness and performance depend on the relative speeds of processors, accuracy of local clocks, or communication delay bounds. Such algorithms arise, for example, in traditional and wireless communications, networks of mobile devices, and shared-memory multiprocessors. The need to prove rigorous theoretical results about timed distributed algorithms makes it important to have a suitable mathematical foundation. An important feature of the TIOA framework is its support for decomposing timed system descriptions. In particular, the framework includes a notion of external behavior for a timed I/O automaton, which captures its discrete interactions with its environment. The framework also defines what it means for one TIOA to implement another, based on an inclusion relationship between their external behavior sets, and defines notions of simulations, which provide sufficient conditions for demonstrating implementation relationships. The framework includes a composition operation for TIOAs, which respects external behavior, and a notion of receptiveness, which implies that a TIOA does not block the passage of time. The TIOA framework also defines the notion of a property and what it means for a property to be a safety or a liveness property. It includes results that capture common proof methods for showing that automata satisfy properties. Table of Contents: Introduction / Mathematical Preliminaries / Describing Timed System Behavior / Timed Automata / Operations on Timed Automata / Properties for Timed Automata / Timed I/O Automata / Operations on Timed I/O Automata / Conclusions and Future Work