The idea that all life on earth traces back to a common beginning dates back
at least to Charles Darwin's {\em Origin of Species}. Ever since, biologists
have tried to piece together parts of this `tree of life' based on what we can
observe today: fossils, and the evolutionary signal that is present in the
genomes and phenotypes of different organisms. Mathematics has played a key
role in helping transform genetic data into phylogenetic (evolutionary) trees
and networks. Here, I will explain some of the central concepts and basic
results in phylogenetics, which benefit from several branches of mathematics,
including combinatorics, probability and algebra.Comment: 18 pages, 6 figures (Invited review paper (draft version) for AMM
