Wolfram's elementary 1D cellular automaton with rule 110 (binary 01101110) is computationally universal — it can simulate any Turing machine, proved by Cook (2004).
Wolfram (1983–2002) classified all 256 elementary rules into 4 classes. Rule 110 is Class IV — complex, aperiodic, capable of universal computation. Cook's proof (published 2004 after 10-year embargo) constructed gliders in Rule 110 that simulate cyclic tag systems equivalent to Turing machines.