Despite the considerable industrial importance of filamentous fungi there have been very few attempts to model the complex growth process of these microorganisms. With a new generation of high performance, computerized bioreactors and new analytical techniques it is possible to obtain the necessary experimental data for setting up reliable structured models describing the growth process of filamentous fungi. It is therefore interesting to review the mathematical models described previously in the literature and the experimental data on which these models are built. Only structured models are considered due to the complex metabolism of filamentous fungi and to the natural cellular structuring of the biomass, i.e. the biomass can be divided into different cell types. In order to set up good structured models it is strictly necessary to have a detailed knowledge of the mechanisms underlying the growth process. This involves both biochemical insight and understanding of the interactions between different macromolecules and cytological organelles.