Abstract: A C*-algebra is often considered as a non-commutative space, which is justified by the natural duality between the category of commutative C*-algebras and the category of locally compact, Hausdorff spaces. Via this natural duality, we transfer Lebesgue covering dimension on locally compact, Hausdorff spaces to nuclear dimension on 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 simple separable 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.