This workshop will examine and compare the different natural metrics on Teichmüller space, from the point of view of coarse geometry, dynamics, and differential geometry. The Teichmüller metric, the Weil-Petersson metric, and Thurston's "asymmetric" metric emphasize and reflect different aspects of the geometry of surfaces, and their similarities and differences are a fascinating and subtle object of study. There are interesting generalizations of these metrics to other settings, notably the asymmetric metric on Culler-Vogtmann Outer Space, and the pressure metric on Hitchin representations in higher rank.