The structure to be expected in the immediate neighbourhood of a given vertex, in many of the most widely used models of networks, can be matched to that of a suitably chosen branching process. An example of the geometric growth typical of a branching process is furnished by the early stages of the spread of an epidemic disease, as evidenced in the COVID-19 pandemic. The Bernoulli random graph, in turn, can be interpreted as being formally equivalent to the well-established Reed–Frost epidemic model, and the growth of an epidemic more generally can be related to the underlying network of contacts between individuals. In this chapter, branching processes in discrete time are introduced, and their basic properties established. For the Bienaymé–Galton–Watson’ process, moment formulae, the criticality theorem, the distribution of the total population size (in the sub-critical and critical cases) and the asymptotically geometric growth (in the super-critical case) are addressed. In the multitype analogue of the Bienaymé–Galton–Watson’ process, the convergence of the type distribution in the super-critical case is also presented.