Abstract: In this talk, we present our research works about variational problems of Riemannian functionals on an $n$-dimensional compact Riemmannian manifold $(M,g)$, which include the renormalized volume coefficients functional $\int_M v^{2k}(g)dv_g$, and the Weyl curvature functional $\int_M |W(g)|^{n/2}dv_g$. For a hypersurface in a sphere, we study the generalized Willmore functional and generalized Willmore conjecture. By use of an inequality between the Weyl curvature functional and the generalized Willmore functional, we give some discussions about the Generalized Willmore conjecture for 4-dimensional compact  hypersurfaces in a sphere.