Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: recent results

In the paper we present the new results in the theory of integrable Hamiltonian systems with two degrees of freedom and topological billiards. The results are obtained by the authors, their students, and participants of scientific seminars of the Department of Differential Geometry and Applications, Faculty of Mathematics and Mechanics at Lomonosov Moscow State University.