When the classic continuous finite element methods cannot meet the needs of modern computational techniques such as hp adaptive and hybrid meshes, discontinuous piecewise polynomials are used in the finite element procedures. This presentation will study the finite element methods that use totally discontinuous approximation functions. Discontinuous Galerkin (DG) methods are such kind methods including IPDG methods, LDG methods and HDG methods. DG methods enforce the continuity of the approximation solutions cross elements by either tuning the penalty parameters or introducing additional equations. The weak Galerkin (WG) finite element method provides a framework for handling discontinuous functions. This general framework will provide a platform for deriving new methods and simplifying the existing methods. A successful examples will be discussed in the presentation.