The notion of a fractal group was introduced recently. Fractal groups are closely related to branch groups that also were defined a few years ago and to groups generated by finite automata which are much older. The study of limit sets of fractal groups and of the spectrum of Markov, Laplace and Hecke type operators related to them leads to the appearence of sets of fractal nature. In particular, the spectra of Hecke type operators can be Julia sets of quadratic maps. Moreover, the fractal groups appear as monodromy groups in holomorphic dynamics. The two dimensional dynamics play an important role in all spectral business.