The simplest known Turing-complete system Proved 2004 — Matthew Cook
Wolfram's elementary cellular automata: each cell is 0 or 1; next state depends on left neighbor, self, right neighbor — 8 possible patterns, each mapped to 0 or 1. That gives 28=256 rules (named by their binary output).
Rule 110 was proved Turing-complete by Matthew Cook (2004, using Wolfram's work). It supports persistent "gliders" and glider guns analogous to Conway's Life — but in 1D! The proof encodes a cyclic tag system using collisions between propagating patterns. Click the canvas to flip cells in the current row and watch patterns emerge. Rule 30 is used in Mathematica's random number generator.