Let $M$ be a closed smooth manifold. Consider the space of Riemannian metrics $\mathcal{M}$ on $M$. A real valued function on $\mathcal
 {M}$ is called a Riemannian functional if it remains invariant under the action of the group of diffeomorphisms of $M$ on $\mathcal{M}$. We will discuss
 some geometric properties of the critical points of certain natural Riemannian functionals in this lecture.