The removal of infectious biofilms from tissues or implanted devices and their transmission through fluid transport systems depends in part of the mechanical properties of their polymeric matrix. Linking the various physical and chemical microscopic interactions to macroscopic deformation and failure modes promises to unveil design principles for novel therapeutic strategies targeting biofilm eradication, and provide a predictive capability to accelerate the development of devices, water lines etc. that minimise microbial dispersal. Here our current understanding of biofilm mechanics is appraised from the perspective of biophysics, with an emphasis on constitutive modelling that has been highly successful in soft matter. Fitting rheometric data to viscoelastic models has quantified linear and non-linear stress relaxation mechanisms, how they vary between species and environments, and how candidate chemical treatments alter the mechanical response. The rich interplay between growth, mechanics and hydrodynamics is just becoming amenable to computational modelling and promises to provide unprecedented characterisation of infectious biofilms in their native state.