The objective of this work is to investigate a Discontinuous Galerkin (DG) method for compressible Euler equations, based on an isogeometric formulation: the partial differential equations governing the flow are solved on rational parametric elements, that preserve exactly the geometry of boundaries defined by Non-Uniform Rational B-Splines (NURBS), while the same rational approximation space is adopted for the solution. We propose a new approach to construct a DG-compliant computational domain based on NURBS boundaries and examine the resulting modifications that occur in the DG method. Some two-dimensional test- cases with analytical solutions are considered to assess the accuracy and illustrate the capabilities of the proposed approach. The critical role of boundary curvature is especially investigated. Finally, a shock capturing strategy based on artificial viscosity and local refinement is adapted to this isogeometric context and is demonstrated for a transonic flow.