Applications of the quantum dilogarithm

The quantum dilogarithm is a special function of two variables that finds various applications, including quantum topology and lattice integrable models of quantum field theory and statistical mechanics. Although a special case of that function was introduced in 1886 by Hölder, its deep connections to quantum world were revealed only in early 1990's after the discovery of the quantum five term identity by Ludwig Faddeev. I will review some of the properties of the quantum dilogarithm, its generalizations to the context of locally compact Abelian groups, and applications in spectral theory, quantum integrable systems and quantum topology.