This work aims to familiarize the reader with essential properties of the chaotic dynamics of Hamiltonian systems, avoiding special mathematical tools not typical for physics. It includes material on separatix chaos, small nonlinearity chaos, fractional kinetics, and discussions on Maxwell's demon and the foundation of statistical physics. The book should be of use to those who are actively working on the problem of dynamical chaos. It introduces the physicist to the world of Hamiltonian chaos and the mathematics to actual physical problems. The material can also be used by graduate students.