In this chapter, we consider KCM with completely arbitrary update families. The goal of universality is to identify what kinds of behaviour are possible within this vast collection of models and to efficiently classify all update families with respect to their behaviour. We begin by considering one-dimensional models, which feature three rough universality classes, represented by the three nearest neighbour models studied in Chap.  4 . We then provide some background on the two-dimensional BP universality theory, before moving on to two-dimensional KCM. We present the complete rough and refined universality theory for KCM in two dimensions and cover the essential elements of their proofs, building on the techniques exhibited in Chaps.  4 and  5 .