Discontinuous Galerkin (DG) methods are a left; LINE-HEIGHT: normal; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" dir=ltr methods have the flexibility which is not shared by typical finite element methods . A left; LINE-HEIGHT: normal; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" dir=ltr The idea of LDG methods is to suitably rewrite a higher order PDE into a first order system , then apply the DG method to the system . A key ingredient for the success of such methods is the correct design of interface numerical fluxes . These fluxes must be designed to guarantee stability and local solvability of all of the auxiliary variables introduced to approximate the derivatives of the solution . In this thesis at first , we introduce some notation and useful subjects for definition of DG mehods , then we present the basic formulation of the discontinuous Galerkin method for scalar conservation laws . We implement a LDG method for the fractional diffusion problem characterized by having fractional derivatives , parameterized by \\beta \\in [1,2] . We show through analysis that one can construct a numerical flux which results in a scheme that exhibit optimal order of convergence O(h^{k+1}) in the continuous range between pure advection (\\beta=1) and pure diffusion (\\beta=2) and also we implement LDG methods for the fractional convection-diffusion with a fractional operator of order \\alpha (1 \\alpha 2) defined through the fractional Laplacian . The fractional operator of order \\alpha is expressed as a composite of first order derivatives and fractional integrals of order 2-\\alpha , and the fractional convection-diffusion problem is expressed as a system of low order differential/integral equations and a local discontinuous Galerkin method scheme is derived for the equations . We also investigate the numerical stability and convergence of the methods.