Abstract: Deformation quantization or a star product ? is a way of producing a noncommutative algebra from the algebra C1(M) of smooth functions on a manifold. A deformation quantization gives rise to a Poisson structure on M : Conversely, which is a very nontrivial result, Kontsevich proved that every Poisson manifold (M; f; g) admits a deformation quantization. The existence of the star product creates an immense collection of examples of noncommu- tative spaces; moreover, they arise from the classical ones! In 2011, Fathizadeh and Khalkhali, and independently Connes and Moscovich in two coauthored papers computed using spectral methods the curavature of the noncommutative tori A . The rst incidence in which the curvature of a noncommutative space had ever been computed by this method. This is achieved by evaluating the value of the analytic continuation of the spectral zeta function. The modular automorphism group from the theory of type III factors and quantum statistical mechanics appears in the nal formula for the curvature. The main instrument here is the asymptotic expansion of the heat trace of the Laplacian in the spectral triple attached to A . In my talk, I will explain the basics in NCG and the recent development respecting my thesis project. Everyone is welcome :)