Universal Turing machine

Remarks

The machine $U$ keeps the description of $M$ on its tape, and the current tape and state of $M$ beside it. At each step it looks up the matching rule of $M$ in the description and updates the simulated configuration. In terms of computable functions, it computes the universal function $(e, x) \mapsto \varphi_e(x)$, which is itself partial computable. Universal machines are the theoretical model of the stored-program computer: one fixed machine runs any program given as data.