ABSTRACT: Elliptic equations have always been an important tool in the study of problems in geometry. In the recent decades, non-linear second order elliptic equations with critical exponents have played a special role in the solutions of several important problems in conformal geometry; e.g. the problem of prescribing Gaussian curvature and the Yamabe problem.

In this talk, I will describe some recent effort to extend the role played by second order equations to higher order ones. First, I will describe properties of a class of conformal covariant operators-- in particular a 4-th order operator with its leading symbol the bi-Laplace operator, discovered by Paneitz in 1983 --; then I will describe the relations of these operators to some natural functionals (e.g. the zeta-functional determinant for the Laplace operator) and elliptic equations (e.g. the Monge-Ampere equation). I will discuss questions of existence, uniqueness and regularity of the associated nonlinear equations.

As applications, I will describe a problem on prescribing the Ricci curvature on 4-dimensional manifolds; and an extension of the Cohn-Vossen inequality to 4-dimension, relating the Euler number to the total integral of a 4-th order curvature invariant.