There have been many fruitful attempts to define noncommutative versions of the covering dimension of a topological space, ranging from the stable and real ranks to the decomposition rank. In 2010, Winter and Zacharias defined the nuclear dimension of a C*-algebra, which has turned out to be a major development in the study of nuclear C*-algebras. In this talk, we introduce nuclear dimension, discuss the differences between it and other dimension theories, and focus on why

nuclear dimension is so important.

(This is a practice for a talk I'm giving at Penn State, so it will be more formal than usual.)