Finite automata and regular languages have been useful in a wide variety of problems in computing, communication and control, including formal modeling and verification. Traditional automata do not admit an explicit modeling of time, and consequently, timed automata [2] were introduced as a formal notation to model the behavior of real-time systems. Timed automata accept timed languages consisting of sequences of events tagged with their occurrence times. Over the years, the formalism has been extensively studied leading to many results establishing connections to circuits and logic, and much progress has been made in developing verification algorithms, heuristics, and tools. This paper provides a survey of the theoretical results concerning decision problems of reachability, language inclusion and language equivalence for timed automata and its variants, with some new proofs and comparisons. We conclude with a discussion of some open problems.