In this project we address the numerical approximation of hyperbolic equations and systems using the discontinuous Galerkin (DG) method in combination with higher order polynomial degrees. In short, this is called Spectral Discontinuous Galerkin (SDG) method. Our interest is to review the theoretical properties of the SDG-method, particularly for what concerns stability, convergence, dissipation and dispersion. Special emphases will be shed on the role of the two parameters,(the grid-size) and (the local polynomial degree). In this respect, we will carefully analyse the available theoretical results from the literature, then we extend some of them and implement several test cases with the purpose of assessing quantitatively the predicted theoretical properties.