Random walks on groups is a part of the theory of Markov processes related to representation theory, invariant means theory, operator algebra theory, asymptotic group theory etc. Among different topics in the study of the asymptotic behaviour of random walks is the spectral direction which deals with the spectrum and the spectral measure of the discrete Laplace operator. After a quick introduction into the subject we shall focus on the case of the lamplighter group and explain how the computation of the spectrum and of the spectral measure was provided in joint work with A. Zuk. The spectrum of the Laplace operator happened to be pure point spectrum and the spectral measure turned out to be discrete. This is the first example of a group with a nontrivial discrete component in spectral measure. The possibility of computation is based on a realization of the lamplighter group as a group generated by a 2-state automaton. As an unexpected application (found in collaboration with P. Linnell, T. Schick and A. Zuk) is a construction of a 7-dimensional closed manifold with noninteger third L²-Betti number. This answers one of M. Atiyah's questions.