Population genetics (the theory of biological evolution at the level of individual genotypes within populations) today relies crucially on mathematical concepts such as ancestral lineages and random genealogies. The talk will provide an overview. It starts from an interacting particle system that describes the joint action of random reproduction, mutation, and selection; then explains the constructions used to trace back the ancestry of individuals and their genealogy, and presents some recent results. In particular, it presents a novel approach that unifies the two established concepts in the field, namely, the ancestral selection graph of Krone and Neuhauser (1997) and the lookdown construction of Donnelly and Kurtz (1999). The talk will be addressed to a general mathematical audience. No previous knowledge of biology is required, nor is specialised knowledge of probability theory.