Newly developed weak Galerkin finite element methods will be introduced for solving partial differential equations. Like discontinuous Galerkin methods, weak Galerkin finite element methods allow to use discontinuous functions in finite element procedure which makes weak Galerkin methods highly flexible. Unlike DG methods, weak Galerkin finite element methods enforce weak continuity of variables through well defined discrete differential operators. Therefore, weak Galerkin methods are parameter independent and absolutely stable. Error analysis and numerical experiments are presented.