First, for nonsymmetric or indefine linear elliptic PDEs, we obtain the first error estimate in L2-norm for the classical two-grid method, then design and analysis an improved two-grid method by adding one more correction on the coarse space to the classical two-gird method. Secondly, for second order semilinear elliptic PDEs, we design and analyze a new conforming finite element discretization technique based on iterative two-grid methods, we also present the DG methods and discuss the corresponding iterative two-grid algorithm. At last, we develop several two-grid methods and two-level additive preconditioners for the Nedelec edge finite element approximation of the time-harmonic Maxwell equations.