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We analyze simulation relations as heuristics for the simplification of omega-automata, i.e., of finite automata working on infinite strings. Our focus is on alternating omega-automata, especially automata resulting from LTL formulas, as is important in model checking. We introduce direct, delayed, and fair simulation for alternating Büchi automata (ABA), and simulation quotient constructions for ABA. Our simulations are compatible with the standard translation of ABA to non-alternating automata. We apply these results to translating formulas of propositional LTL to nondeterministic automata. We develop a translation algorithm from LTL to nondeterministic automata with an on-the-fly use of simulation relations for simplification, and we compare our approach to tableau-based translation algorithms. We extend our notion of delayed simulation to alternating parity automata (APA), introduce variants of this relation suited for quotienting of APA, and develop a simulation-based simplification algorithm for APA. We give a sketch of how to apply these results to a fragment of the modal mu- calculus. This book is suited for students and researchers interested in the automata theory of LTL.