The article surveys recent results in the study of topological complexity of recognizable tree languages. Emphasis is put on the relation between topological hierarchies, like the Borel hierarchy or the Wadge hierarchy, and the hierarchies resulting from the structure of automata, as the Rabin-Mostowski index hierarchy. The topological complexity of recognizable tree languages is seen as an evidenceof their structural complexity, which also induces the computational complexity of the verification problems related to automata, as the non-emptiness problem. Indeed, the topological aspect can be seen as a rudiment of the infinite computation complexity theory.