Consider the space of maps between two nice spaces X and Y . Traditionally, homotopy theory might be concerned with classifying the connected components of this mapping space. In the 1990's, Gromov pointed out that this space has extra structure coming from metrics on the spaces: we might be interested not only in the existence of a map or of a homotopy between two maps, but how "efficient" it is. I will discuss the motivations for this program, some old observations and conjectures due to Gromov, as well as some recent results.