The biodiversity that we observe on Earth is ultimately the product of evolutionary forces: natural selection, mutations, recombination, and random genetic drift jointly shaped the genome of all living organisms. By employing Next Generation Sequencing techniques, geneticists are now able to observe patterns of DNA sequence variation at different evolutionary scale and with an unprecedented level of details. Coalescent theory represent arguably the most robust mathematical framework describing the connection between those changes in the genome and the demographic history of a species. In this seminar, I will introduce the standard coalescent model, how this arises from Poisson processes and its connection with a simple population genetic model (the Wright-Fisher model). I will then discuss how can we build more sophisticated models within the coalescent framework and how we can apply them to reconstruct the evolutionary history of a species and redefine the species concept itself.