Recently, there have been a breakthrough in understanding the geometry of curved noncommutative tori; The scalar curvature is introduced and explicitly computed for curved noncommutative tori through special values of spectral zeta functions. The focus of this talk will be on the Ricci curvature, which is a more subtle invariant of Riemannian geometry. We shall give a spectral formulation of Ricci curvature in the case of closed Riemannian manifolds in terms of residues and poles of spectral zeta functions. Then we apply this formulation to the noncommutative two torus and show how Connes' pseudodifferential calculus can be used to compute the Ricci density explicitly for these noncommutative spaces.