A central pursuit in biology is a description of the demographic aspects of the processes, which have led to the existence of present-day populations as we encounter them in nature. An approach to this endeavour is provided by recent advances in DNA sequencing technology, which have made available a large number of new alignments of whole-genome sequence data.

Generalizing classical coalescent theory, we develop mathematical models whose parameters are interpreted as the quantities of interest.These parameters include split times, where ancestral populations split into descendant populations, migration rates between co-existing populations and population sizes. In this talk I will discuss such mathematical models, with the so-called Isolation With Migration model as a point of departure. I will describe how to formulate such models in a Continuous Time Markov Chain framework, and I will show how to address the issue of state-space explosion which one encounters, using the concept of lumpability. Furthermore, I will show how to model different assumptions about mutations,and how to incorporate these assumptions into the model. In particular, I will describe how an  infinitely-many alleles assumption may be modelled and show how this leads to a generalization to multiple populations, of the celebrated Ewens Sampling formula.