  We analyze nonlinear degenerate coupled PDE-PDE and PDE-ODE systems that
arise, for example, in the modelling of biofilm growth. One of the equations,
describing the evolution of a biomass density, exhibits degenerate and singular
diffusion. The other equations are either of advection-reaction-diffusion type
or ordinary differential equations. Under very general assumptions, the
existence of weak solutions is proven by considering regularized systems,
deriving uniform bounds, and using fixed point arguments. Assuming additional
structural assumptions we also prove the uniqueness of solutions.
  Global-in-time well-posedness is established for Dirichlet and mixed boundary
conditions, whereas, only local well-posedness can be shown for homogeneous
Neumann boundary conditions. Using a suitable barrier function and comparison
theorems we formulate sufficient conditions for finite-time blow-up or uniform
boundedness of solutions. Finally, we show that solutions of the degenerate
parabolic equation inherit additional global spatial regularity if the
diffusion coefficient has a power-law growth.
