This book delves into a rapidly evolving area of classical mechanics: the theory of integrating Hamiltonian equations. For the first time, it presents a systematic analysis of the reasons behind the non-integrable behavior of Hamiltonian systems. Key topics include the intricate structure of phase space, small denominators, the splitting of asymptotic surfaces, the emergence of isolated periodic solutions, branching of solutions in the complex time plane, and quasi-random oscillation modes. The text outlines methods for integrating Hamiltonian systems and lists numerous precisely solved problems. General results are illustrated with examples from celestial mechanics, rigid body dynamics, hydrodynamics, and mathematical physics. This book is intended for specialists in mechanics and mathematics who are engaged in the theory of dynamic systems, as well as for university students and graduate researchers.