The basic idea behind fractional calculus has a history during three hundred years that is similar to and aligned with that of more classical calculus and the topic has attracted the interests of mathematicians who contributed fundamentally to the development of classical calculus, including L’Hospital, Leibniz, Liouville, Riemann, Grünward, and Letnikov. In spite of this, the development and analysis of fractional calculus and fractional differential equations are not as mature as that associated with classical calculus. However, during the last decade this has begun to change as it has become clear that fractional calculus naturally emerges as a model for a broad range of non-classical phenomena in the applied sciences and engineering. A striking example of this can be considered as a model for anomalous traort processes and diffusion, leading to partial differential equations (PDEs) of fractional type. These models are found in a wide range of applications such as porous flows, models of a variety of biological processes, and traort in fusion plasmas, to name a few. With this emerging range of applications and models based on fractional calculus, a need for the development of robust and accurate computational methods for solving these equations will be necessary. A fundamental difference between problems in classical calculus and fractional calculus is the global nature of the latter formulations. Nevertheless, methods based on finite difference methods and finite element formulations have been developed and successfully applied. The discontinuous Galerkin (DG) finite element method is a very attractive method for solving PDEs because of its flexibility and efficiency in terms of mesh and shape functions, and the higher order of convergence can be achieved without over many iterations. The discontinuous Galerkin method is a well established method for classical conservation laws. However, for equations containing higher order spatial derivatives, discontinuous Galerkin methods cannot be directly applied.