It is well known that certain systems in classical mechanics are completely integrable in a sense that they possess a sufficient number of integrals of motion which allow us to integrate equations of motion explicitly. In Hamiltonian formulation the number of integrals of motion should be no less than the number of canonical coordinates (or canonical momenta). Then there exist special canonical variables called action-angle variables such that the dynamics of the system in these coordinates reduces to a set of one-dimensional problems.
These ideas can be extended to quantum integrable systems. In quantum case we are looking for a complete set of eigenfunctions which simultaneously diagonalize ``quantum integrals of motion''. Quantum separation of variables allows us to express these multi-variable eigenfunctions in terms of functions of one variable only. In principle this helps to solve the equations of motions for the original system. Surprisingly such (integral) transformations are linked to some well known (or sometimes new) formulas in the theory of classical special functions.