Abstract: The study of higher order energy functionals was first proposed by Eells and Sampson in 1965 and, later, by Eells and Lemaire in 1983. These functionals provide a natural generalization of the classical energy functional. In the first part of the lecture we will give an account on two different higher order energy functionals that have been studied in the literature. The first is the original functional proposed by Eells-Sampson that called the ES-r-energy, the other, proposed and studied by Maeta (and others), is simply called r-energy. We will then give examples of proper critical points of the ES-r-energy and also show some general facts which should motivate future developments of this subject. In the second part of the talk we will consider hupersurfaces in space forms whose inclusion map is a critical point of the r-energy functional. For this class of hypersurfaces we will show that, if c? 0, then any r-harmonic hypersurface must be minimal provided that the mean curvature function and the squared norm of the shape operator are constant. When the ambient space is the m-dimensional sphere, we will obtain the geometric condition which characterizes the r-harmonic hypersurfaces with constant mean curvature and constant squared norm of the shape operator, and we will establish the bounds for these two constants.