Population movements over space and time played a crucial role in generating the genetic patterns that are observed in the present day. Numerous factors, such as climate changes or cultural innovations, have the potential to induce large-scale movements, such as population expansions (i.e. increases both in density and range) or contractions to refugee areas. It is thus very important to take the spatial dynamic of populations into account when trying to reconstruct their history from genetic data. Computer simulation constitutes a very powerful tool for the study of the combined impacts of biological and demographic factors on the genetic structure of populations. The rapid increase of computer power opens many new possibilities for research in that specific area. A series of recent studies have focused on the consequences of population expansions on their genetic diversity. These studies extensively described one potentially important genetic process which may occur during a range expansion: the "mutation surfing" phenomenon. In this paper, we describe in detail this process and its potential implications for the establishment of the current genetic diversity in Europe. We also discuss the limitations and perspectives of such computer simulation studies in the field, and possible future improvements to them.