The paper introduces the concept of alternation, a generalization of nondeterminism where existential and universal quantifiers can alternate during computation. Alternating Turing machines (ATMs) are defined and shown to accept exactly the recursively enumerable sets. The paper characterizes the complexity classes of languages accepted by time- and space-bounded ATMs in terms of deterministic complexity classes. Specifically, alternating polynomial time is equivalent to deterministic polynomial space, and alternating polynomial space is equivalent to deterministic exponential time. The paper also defines subrecursive quantifier hierarchies based on bounded alternation and shows that alternating finite automata accept only regular languages, with $2^{2^k}$ states being necessary and sufficient for deterministic simulation. Finally, it demonstrates that alternating pushdown automata are strictly more powerful than nondeterministic pushdown automata. The paper discusses the implications of alternation in various areas of computer science, including time and space complexity, logic, games, and parallelism.The paper introduces the concept of alternation, a generalization of nondeterminism where existential and universal quantifiers can alternate during computation. Alternating Turing machines (ATMs) are defined and shown to accept exactly the recursively enumerable sets. The paper characterizes the complexity classes of languages accepted by time- and space-bounded ATMs in terms of deterministic complexity classes. Specifically, alternating polynomial time is equivalent to deterministic polynomial space, and alternating polynomial space is equivalent to deterministic exponential time. The paper also defines subrecursive quantifier hierarchies based on bounded alternation and shows that alternating finite automata accept only regular languages, with $2^{2^k}$ states being necessary and sufficient for deterministic simulation. Finally, it demonstrates that alternating pushdown automata are strictly more powerful than nondeterministic pushdown automata. The paper discusses the implications of alternation in various areas of computer science, including time and space complexity, logic, games, and parallelism.