Branching process theory provides appropriate mathematical models to describe the probabilistic evolution of systems whose components (cells, particles, individuals in general) after a certain life period reproduce and die. The genealogy of this theory is presented, describing the basic results of the standard Bienaymé-Galton-Watson process and some of its generalizations, paying special attention to controlled branching processes. Several fields in which this theory has been applied successfully are indicated.

Keywords: Branching processes, extinction probability, asymptotic distribution, history, applications.

AMS Subject classifications: 60J80, 62M05.