We survey the effect of perturbing the regular structure of a cellular automaton. We are interested in critical phenomena, i.e., when a continuous variation in the local rules of a cellular automaton triggers a qualitative change of its global behaviour. We focus on three types of perturbations: (a) when the updating is made asynchronous, (b) when the transition rule is made stochastic, (c) when the topological defects are introduced. It is shown that although these perturbations have various effects on CA models, they generally produce the same effects, which are identified as first-order or second-order phase transitions. We present open questions related to this topic and discuss implications on the activity of modelling.