The notion of a just infinite group leads naturally to the concept of a branch group (as one of the possible three types of just infinite groups), which then leads to the notion of a finitely constrained group (analog of a shift of finite type in symbolic dynamics). Along the way, we enter into discussion of pro-N aspects of self-similar groups for various families N of normal subgroups (pro-finite and pro-p completions, congruence property, closures, Hausdorff dimension, etc.).