  We consider percolation on high-dimensional product graphs, where the base
graphs are regular and of bounded order. In the subcritical regime, we show
that typically the largest component is of order logarithmic in the number of
vertices. In the supercritical regime, our main result recovers the sharp
asymptotic of the order of the largest component, and shows that all the other
components are typically of order logarithmic in the number of vertices. In
particular, we show that this phase transition is quantitatively similar to the
one of the binomial random graph.
  This generalises the results of Ajtai, Koml\'os, and Szemer\'edi and of
Bollob\'as, Kohayakawa, and \L{}uczak who showed that the $d$-dimensional
hypercube, which is the $d$-fold Cartesian product of an edge, undergoes a
phase transition quantitatively similar to the one of the binomial random
graph.
