This volume describes and fully illustrates both the theory and applications of integrable Hamiltonian systems. Exploring the basic elements of Liouville functions and their singularities, it systematically classifies such systems for the case of integrable Hamiltonian systems with two degrees of freedom. It also describes the nontrivial connections between this theory and three-dimensional topology and gives a topological description of the behavior of integral trajectories under Liouville tori bifurcation. Integrable Hamiltonian Systems: Geometry, Topology, Classification will appeal to graduate students of mathematics and mathematicians working in the theory of dynamical systems and their applications.