In this review, we discuss applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers. We further describe how empirical data, e.g. distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birth-death-and-innovation models. We conclude that birth-and-death processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological processes, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.