The Dirichlet biharmonic equation occurs in many areas of science and engineering, including fluid mechanics, elasticity, material science, etc. It is a fourth order partial differential equation (PDE) which means that the numerical solution of this equation is far more difficult than second order PDEs such as the Poisson equation. We shall use the preconditioned conjugate method to solve the finite element problem in a complexity proportional to the number of unknowns, provided an optimal Poisson solver is available. The crucial step is to find a preconditioner based on the Poincare-Steklov operator (or Dirichlet to Neumann map) for a pseudodifferential operator. This method works for smooth domains in any number of space dimensions. It builds upon the fundamental work by Glowinski and Pironneau. Extensions to self-adjoint elliptic and parabolic fourth order PDEs will also be given.