I will begin by giving an introduction to self-similar groups and their role in modern mathematics. This will in particular require appearance of finite automata and automatically generated sequences, Cantor systems and minimal sushifts of the Bernoulli shift.
Then I will explain the role of Bernoulli measures in the dynamics of self-similar groups and show that their images under the automaton maps are self-similar measures.
Finally, I will discuss the connections between spectral theory of discrete Laplacians and random Schrodinger operators. I will explain the role of multidimensional rational dynamics in the spectral problem on Cayley and Schreier graphs, discuss the question "Can one hear the shape of group?" and report on new results on spectra of self-similar groups and their Schreier graphs.
Such well-known examples of self-similar groups as the lamplighter group, Basilica group, Hanoi towers group, iterated monodromy groups and groups of intermediate growth will be in the center of our attention.