In noncommutative geometry, the geometry of a space is given via a spectral triple $(\mathcal{A,H},D)$. Geometric information, in this approach, is encoded in the spectrum of $D$ and to extract them, one should study spectral functions such as the heat trace $\Tr (e^{-tD^2})$, the spectral zeta function $\Tr(|D|^{-s})$ and the spectral action functional, $\Tr f(D/\Lambda)$. The main focus of this thesis is on the methods and tools that can be used to extract the spectral information. Applying the pseudodifferential calculus and the heat trace techniques, in addition to computing the newer terms, we prove the rationality of the spectral action of the Robertson-Walker metrics, which was conjectured by Chamseddine and Connes. In the second part, we define the canonical trace for Connes' pseudodifferential calculus on the noncommutative torus and use it to compute the curvature of the determinant line bundle for the noncommutative torus. In the last chapter, the Euler-Maclaurin summation formula is used to compute the spectral action of a Dirac operator (with torsion) on the Berger spheres $\mathbb{S}^3(T)$.