Abstract: In a monotone cellular automaton, each site in the d-dimensional integer lattice can at each integer time take the values zero or one. The value of a site at a given time is a monotone function of the values of the site and finitely many of its neighbours at the previous time. Toom’s stability theorem gives necessary and sufficient conditions for the all one state to be stable under small random perturbations. We review Toom’s Peierls argument and extend it to random cellular automata, in which the functions that determine the value at a given space-time point are random and i.i.d. We are especially interested in the case where with positive probability, the identity map is applied, that just copies the value of a site at the previous time. We derive sufficient conditions for the stability of such random cellular automata. Joint work with Cristina Toninelli and Jan Swart.