\(C^0\) interior penalty methods are discontinuous Galerkin methods for fourth order problems. In this talk I will discuss the advantages of \(C^0\) interior penalty methods over classical finite element methods and introduce a framework for their analysis that is applicable to both fourth order elliptic boundary value problems and elliptic variational inequalities. I will present the results of two applications of \(C^0\) interior penalty methods to nonlinear problems: the von Karman model for plate buckling and the obstacle problem for the bending of Kirchhoff plates.