Alternation is a generalization of nondeterminism in which existential and universal quantifiers can alternate during computation. Alternating Turing machines (ATMs) accept precisely the recursively enumerable sets. Complexity classes of languages accepted by time- or space-bounded ATMs are characterized in terms of deterministic complexity classes. Alternating polynomial time is equivalent to deterministic polynomial space, and alternating polynomial space is equivalent to deterministic exponential time. Subrecursive quantifier hierarchies are defined using bounded alternation. Alternating finite-state automata accept only regular languages, though they may require exponentially more states than nondeterministic ones. Alternating pushdown automata are strictly more powerful than nondeterministic pushdown automata. Alternation has applications in complexity theory, logic, and parallelism, enabling succinct representations and increased computational power in certain cases. The paper defines ATMs, shows their equivalence to recursively enumerable sets, and characterizes their complexity classes. It also discusses alternating automata and their relationships to deterministic and nondeterministic counterparts.Alternation is a generalization of nondeterminism in which existential and universal quantifiers can alternate during computation. Alternating Turing machines (ATMs) accept precisely the recursively enumerable sets. Complexity classes of languages accepted by time- or space-bounded ATMs are characterized in terms of deterministic complexity classes. Alternating polynomial time is equivalent to deterministic polynomial space, and alternating polynomial space is equivalent to deterministic exponential time. Subrecursive quantifier hierarchies are defined using bounded alternation. Alternating finite-state automata accept only regular languages, though they may require exponentially more states than nondeterministic ones. Alternating pushdown automata are strictly more powerful than nondeterministic pushdown automata. Alternation has applications in complexity theory, logic, and parallelism, enabling succinct representations and increased computational power in certain cases. The paper defines ATMs, shows their equivalence to recursively enumerable sets, and characterizes their complexity classes. It also discusses alternating automata and their relationships to deterministic and nondeterministic counterparts.