We propose a Discontinuous Galerkin (DG) method for the numerical solution of the compressible Navier-Stokes equations with a functional representation derived from Computer Aided Design (CAD). The extraction of a set of DG-compliant basis functions from Non-Uniform Rational B-Splines (NURBS) is briefly discussed and the discretization of the equations is presented in this new framework. The benefits of using high-order geometries are assessed in a first numerical example. The proposed scheme is then extended to deformable domains through an Arbitrary Lagrangian-Eulerian (ALE) formulation and a NURBS-based mesh movement, allowing smooth high-order deformations. A verification test case is carried out to evaluate the impact of mesh movement on numerical solutions. Finally, we propose a dynamic mesh deformation-adaptation algorithm to couple the ALE formulation with Adaptive Mesh Refinement (AMR), based on hierarchical properties of NURBS. The potential of the developed approach is investigated in an application dealing with a pitching aifoil.