Biofilms are dense aggregations of microbial cells encased in a slimy extracellular matrix that form on surfaces in moist environments. They play an important role in various natural and industrial systems: e.g., biofilms are beneficially used in environmental engineering technologies for groundwater protection or wastewater treatment, they can form on implants or natural surfaces in the human body and provoke bacterial infections, or biofouling and biocorrosion of industrial equipment can cause severe economic defects for the industry. We discuss deterministic continuum models for the growth of spatially heterogeneous multi-species biofilms that extend an earlier prototype single-species growth model. While the dissolved substrates typically satisfy classical semilinear reaction-diffusion equations, the governing equations for the biomass fractions exhibit two non-linear diffusion effects: a power-law degeneracy (like the porous medium equation) and super diffusion. Very recent multi-species extensions include cross-diffusion effects and hence, allow to take into account that the movement of one species is affected by the presence of the others.