This is a concise survey of an important area of Dynamical systems, which has its origins in the work of H. Poincare. The notion of asymptotic cycle generalizes the notion of rotation number for orientation preserving homeomorphisms of the circle to dynamical systems on any compact metric space. It is written by the founder of the theory of asymptotic cycles, who in the late 50's made the most important advance in the theory of rotation numbers after Poincare. The article is concentrated on the basic ideas of the theory, so that the reader will not be lost in details. These can be found in the bibliography, which to my knowledge is almost exhaustive.