Biofilms represent the preferential microbial lifestyle and carry out a wide range of reactions which are relevant to many disciplines, such as environmental engineering, food processing, and medicine. Here we discuss a continuum approach to mathematical modeling of multispecies biofilm formation and growth, which usually leads to consider free boundary value problems for nonlinear PDEs. The emphasis will be on the qualitative analysis, uniqueness and existence of solutions and their main properties. Special attention will be also given to the mathematical modelling of the invasion and dispersal phenomena. Numerical applications to real cases of biological and engineering interest will be presented.