In Chap. 4 , the characterization of CTMCs and their limiting and stationary distributions were discussed in detail. Birth-Death ProcessesBirth-death processes (BDPs), whose state space is the state of non-negative integers, are a subclass of Markov chains with continuous-time parameters. These processes are distinguished by the fact that if a transition happens, it leads to a neighbouring state. When a stochastic process consists of the absorption and emission of photons or particles, the excitation and de-excitation of atoms or nuclei, or of electrons in semiconductors, the birth and death of individuals, the arrival and departure of customers, it is characterized by birth and death (one-step or generation-recombination) processes. In this chapter, a detailed study on BDPs is presented. Further, some special BDPs such as pure birth and pure death processes, are also discussed. Also another important stochastic process, namely Poisson process, the basic concepts and properties of the Poisson process both time-homogeneous and in-homogeneous have been provided. Finally, a section on continuous-time branching processes, in particular, the Bellman-Harris branching process is presented.