Three research directions in non-uniform cellular automata

The paper deals with recent developments about non-uniform cellular automata. After reviewing known results about structural stability we complete them by showing that also sensitivity to initial conditions is not structurally stable. The second part of the paper reports the complexity results about the main dynamical properties. Some proofs are shortened and clarified. The third part is completely new and starts the exploration of the fixed points set of non-uniform cellular automata.