A C*-algebra is often considered as non-commutative space, which is justified by the natural duality between the category of unital, commutative C*-algebras and the category of compact, Hausdorff spaces. Via this natural duality, we transfer Lebesgue covering dimension on compact, Hausdorff spaces to nuclear dimension on unital, commutative C*-algebras. The notion of nuclear dimension for C*-algebras was first introduced by Winter and Zacharias, and it has come to play a central role in the structure and classification for simple nuclear C*-algebras. Indeed, after several decades of work, one of the major achievements in C*-algebra theory was completed: the classification via the Elliott invariant for non-elementary simple separable unital C*-algebras with finite nuclear dimension that satisfy the universal coefficient theorem. Unfortunately, simple C*-algebras suffer from a phenomenon of dimension reduction: every simple C*-algebra with finite nuclear dimension must have nuclear dimension at most one. In order to overcome this phenomenon, we (together with Liao and Winter) have introduced the notion of diagonal dimension for an inclusion of C*-algebras, where D is a commutative sub-C*-algebra of A so that this new dimension theory generalizes Lebesgue covering dimension of D and nuclear dimension of A simultaneously. In this talk, I will explain its future impact on the classification of simple nuclear C*-algebras and its connection to dynamic asymptotic dimension introduced by Guentner, Willett and Yu. Note: This talk is part of the activity of the MUR Excellence Department Project MatMod@TOV (CUP E83C23000330006). The speaker is supported in part by INDAM (CUP E53C23001670001) The Operator Algebra Seminar schedule is here: https://sites.google.com/view/oastorvergata/home-page?authuser=0