I offer a leisurely introduction to the 'what', 'why', 'what exactly' and 'how' of my research, which revolves around groups acting on trees. Following a motivation of the subject by situating said groups within the broader theory of all groups, I explain the meaning of 'local' and 'global' in this context. With some examples of groups acting on trees at hand, I illustrate how, in general, 'local' actions have 'global' implications. (Credit to Alejandra Garrido for the title!)

We introduce the notion of self-similarity for groups acting on regular rooted trees as well as their description using automata and wreath iteration. Following the definition of Grigorchuk's group we show that it is an infinite, finitely generated 2-group. The proof illustrates the use of self-similarity.

I'll give an overview of some recent developments in the theory of groups of automorphisms of trees which are discrete in the full automorphism group of the tree and are locally-transitive. I'll also mention some questions which have been provoked by this work.

We generalize the Burger-Mozes universal groups acting on regular trees by prescribing the local action on balls of a given radius, and study the basic properties of this construction. We then apply our results to prove a weak version of the Goldschmidt-Sims conjecture for certain classes of primitive permutation groups.