Mathematical model of computation: an infinite tape over an alphabet, a read/write head that can move left/right, a finite set of states, and a transition function mapping (state, symbol) to (new state, written symbol, direction). More powerful than a real computer because of its infinite tape. Used for mathematical proofs rather than for computing. Key results: one Turing machine can simulate another; determinism does not change expressive power (a deterministic machine can simulate a non-deterministic one, just slower); multitape machines reduce to single-tape machines — the proof that a single-memory computer can exist (historically code and data segments lived separately, e.g. in x86 assembly). A universal Turing machine has two tapes — one for code, one for data — and serves as a mathematical model of a computer.