Automata on infinite words were introduced by J.R. Buchi (1962) in order to give a decision procedure for S1S, the monadic second-order theory of one successor. D.E. Muller (1963) suggested deterministic ω-automata as a means of describing the behavior of nonstabilizing circuits. R. McNaughton (1966) proved that the classes of languages accepted by nondeterministic Buchi automata and by deterministic Muller automata are the same. His construction and its proof are quite complicated, and the blow-up of the construction is doubly exponential. The author presents a determinization construction that is simpler and yields a single exponent upper bound for the general case. This construction is essentially optimal. It can also be used to obtain an improved complementation construction for Buchi automata that is also optimal. Both constructions can be used to improve the complexity of decision procedures that use automata-theoretic techniques.