In this talk we present some recent advances on nonconforming methods for general polyhedral meshes. More specifically, in the first part of the talk we introduce the concept of admissible mesh sequence and establish some functional analytic results on broken polynomial spaces, namely broken Sobolev embeddings and a discrete version of the Rellich-Kondrachov theorem. These tools are then used in the context of discontinuous Galerkin (dG) methods to infer convergence results for problems encountered in subsoil modeling. The focus is on flows through deformable porous media in the elastic regime. In the last part of the talk we propose an unconventional way of reducing the number of degrees of freedom in the context of dG methods, yielding the so-called cell centered Galerkin (ccG) methods. The ideas are applied to the steady Navier-Stokes equations.