A number of recent results, including special discretization schemes, adaptive methods and multilevel iterative methods for the resulting algebraic systems, will be presented in this talk for various partial differential equations (PDEs). With a careful and combined use of qualitative properties of PDEs, the underlying functional spaces and their discretizations, many different kinds of equations will be treated with similar techniques. After an introduction to some practically efficient methods such as the algebraic multigrid method for the Poisson equations, it will be shown how more complicated systems such as linear elasticity equations, electro- magnetic equations, porous media, Stokes equations and more general Newtonian/non-Newtonian models can be reduced to the solution of a sequence of Poisson equation and its simple variants. The efficiency of these algorithms will be illustrated by theoretical analysis, numerical examples and engineering applications.