Abstract: One knows from classical dierential geometry and geometric analysis that several invariants of compact manifolds can be encoded in terms of the spectrum of geometric operators like Laplacian or Dirac operator on a manifold. By a general theorem of Seeley-DeWitt-Gilkey [5], terms in the asymptotic expansion of the heat flow associated to the Laplacian on compact manifolds, contains information like volume (Weyl's law), scalar curvature and more subtle invariants. Some of these techniques can be extended to the setting of noncommutative geometry thanks to the formalism of spectral triples and spectral zeta functions. A recent breakthrough in this area was the work of Connes-Tretko [2] and Fathizadeh-Khalkhali [3] where they proved the Gauss-Bonnet theorem for NC 2-torus. A more recent development is the work of Connes-Moscovici [1] and Fathizadeh-Khalkhali [4] where they have managed to compute the scalar curvature of the NC 2-torus by explicitly computing the value of the zeta functional \zeta_{\Delta}(s) = tr (a\Delta^{-s}). In my presentation I will go through the recent works and their main ideas and also discuss some of those ideas for NC 4-torus and some results which we have recently found for NC 4-tori. References [1] A. Connes, H. Moscovici, Modular curvature for noncommutative two-tori, arXiv:1110.3500. [2] A. Connes, P. Tretko, The Gauss-Bonnet theorem for the noncommutative two torus, arXiv:0910.0188. [3] F. Fathizadeh, M. Khalkhali, The Gauss-Bonnet theorem for noncommuta- tive two tori with a general conformal structure, arXiv:1005.4947. [4] F. Fathizadeh, M. Khalkhali, Scalar curvature for the noncommutative two torus, arXiv:1110.3511. [5] P. Gilkey, Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Mathematics Lecture Series, 11. Publish or Perish, Inc., Wilming- ton, DE, 1984. 1