After Gromov, there is a notion of a distance between two metric spaces which makes the moduli of compact metric spaces itself into a metric space. In this talk I will describe an attempt to define metrics on stacks and to extend Gromov's distance to so-called metric stacks. I will show how metric stacks relate to Rieffel's quantum metric spaces (which are in some sense noncommutative metric spaces), and I will also talk about the applications in physics which motivate the work.