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Noise and random fluctuations play an important role in quantum electronic devices and systems. Such fluctuations reside, for example, in the random creation of photons in optical amplifying media and in the generation of electron-hole pairs in ionizing regions of semiconductor devices. We provide an overview of the fundamental random branching processes that underlie optical and electronic gain. We describe branching processes as concatenations of basic elements that comprise filtered Poisson processes (shot noise) driving secondary Poisson processes. In the presence of feedback, these elementary processes become self-exciting in nature; they are then suitable for characterizing squeezed light and sub-Poisson photon emissions.