The word 'monodromy' means 'once around a course', and it refers to changes that might occur when a system goes around some closed loop [1]. We discuss here a phenomenon whose proper name is 'nontrivial monodromy of action and angle variables in a Hamiltonian system'; for the obvious reason, we just call it 'Hamiltonian monodromy'. In this paper we describe two manifestations of Hamiltonian monodromy: a manifestation in a time-dependent classical system, and a manifestation in a stationary quantum system. Then we give a brief description of the mathematical theory, and finally close with a short survey of previous work on this subject.