Two birth and death processes in continuous time are used in the book. The first is the Yule process, the pure birth process with constant per capita birth rate. The distribution of the number of individuals at any given time is derived, and its asymptotically exponential growth, together with the convergence of the age distribution, are also established. The process is used in the analysis of shortest path lengths in small world processes. The second process is the immigration–death process with constant immigration rate and constant per capita death rate, for which the distribution of the number of individuals at any given time and its distance in total variation from its equilibrium Poisson distribution are investigated. Multitype generalizations are also introduced, in which individuals may have any one of a number of different types. These processes are of purely technical interest and are used in deriving properties of the solutions to the Stein equations associated with the Poisson distribution and its generalizations.