This review focuses on linear birth-and-death processes (LBDPs), describing the basic properties of the population-size process and the underlying ancestral trees that record how the evolving species (or individuals or cells) are related. The first section describes the Yule, or linear birth, process setting. Analogous results for the birth-and-death process (BDP) are given. The stochastic structure of the reconstructed tree obtained by pruning branches that do not survive to the present time is detailed. In §2, the BDP with immigration is described. Immigration is a mechanism to introduce new types into a population evolving through time. For the Yule process, marked Poisson process arguments are used to illustrate properties of the sample variance of the number of families observed in two consecutive time intervals. In the final section, we describe a recent method for approximate Bayesian computation using random forests, and illustrate it with an example of inference from DNA sequence data about the split rate and mutation rate in a birth-and-death model for the evolution of a cell population.This article is part of the theme issue '"A mathematical theory of evolution": phylogenetic models dating back 100 years'.