In my thesis, I formulate, analyze and implement high order discontinuous Galerkin methods for simulating miscible displacement in porous media. The analysis concerning the stability and convergence under the minimal regularity assumption is established to provide theoretical foundations for using discontinuous Galerkin discretization to solve miscible displacement problems. The numerical experiments demonstrate the robustness and accuracy of the proposed methods. The performance study for large scale simulations with highly heterogeneous porous media suggests strong scalability which indicates the efficiency of the numerical algorithm. The simulations performed using the algorithms for physically unstable flow show that higher order methods proposed in thesis are more suitable for simulating such phenomenon than the commonly used cell-center finite volume method.