Self-similarity (when part of an object is a scaled version of the whole) is one of the most basic forms of symmetry. While known and used since ancient times, its use and investigation took off in the 1980s thanks to the advent of fractals, whose infinite self-similar structure has captured the imagination of mathematicians and lay people alike.

Self-similar fractals are highly symmetrical, so much so that even their symmetry groups exhibit self-similarity. In this talk, I will introduce and discuss groups which are self-similar, or fractal, in an algebraic sense; their connections to fractals, symbolic dynamics and automata theory; how they produce fascinating new examples in group theory, and some research questions in this lively new area.