After a quick introduction into amenability and branch groups, we will show how to model the famous Hanoi Towers Game on k>2 pegs by action of a self-similar group H^k (called k-th Hanoi Towers Group)on a rooted k-regular tree. We will discuss properties of these groups related to amenability, growth, branching, and will consider several asymptotic characteristics of associated Schreier graphs including the spectrum, which (in case of three pegs) will be computed by using of the Schur complement trick. A number of open problems will be formulated. (This is based on joint work with V.Nekrashevych and Z.Sunic.)