I will present few results concerning spectral theory of graphs and groups that are based on the use of multidimensional rational maps and self-similar groups and concept of joint spectrum of a tuple of operators. At the beginning there will be a short survey on spectra of infinite graphs and groups. Then I will continue with some results belonging to the speaker and his collaborators (L.Bartholdi, A.Dudko, D.Lenz, T.Nagnibeda, V.Nekrashevych, B.Simanek and Z.Sunic) that are based on the use of self-similar groups. The relation to the random Schrödinger operator will be mentioned at the end of the talk.