Mathematical modeling of spatial biofilm structure has been in development for the past 10 years, its main goal being to derive the dynamics of biofilm structure from first-principle descriptions of the various physical, chemical and biological processes involved in biofilm formation. Early efforts described development of unrestricted monospecies consortia, often considering diffusion and reaction of a single solute species. Multi-dimensional modeling of biofilms has presently reached a stage where multi-species systems with any number of bacterial and solute species, reactions and arbitrary detachment scenarios may be readily implemented using a general-purpose software framework introduced recently. The present work presents motivations for the mathematical modeling of biofilm structure and provides an overview on major contributions to this field from pioneering efforts using cellular automata (CA) to more recent methods using the preferred individual-based modeling (IbM). Recent examples illustrate how biofilm models can be used to study the microbial ecology in: (a) development of multi-species nitrifying biofilms with anammox bacteria, (b) interspecies hydrogen transfer in anaerobic digestion methanogenic consortia, (c) competition between flock-formers and filamentous bacteria influenced by environmental conditions and its effect on morphology of activated sludge flocs, and (d) a two-species biofilm system with structured biomass describing extracellular polymeric substances (EPS) and internal storage compounds. As recent efforts from direct comparison of structure predicted by three-dimensional modeling with that observed by confocal laser scanning microscopy imaging of biofilms grown in laboratory flow cells show a good agreement of predicted structures, multi-dimensional modeling approaches presently constitute a mature and established methodology to enhance our understanding of biofilm systems.