A lecture on "Best Lipschitz Maps" presented by Georgios Daskalopoulos as part of the "New trends in Teichmüller theory" program at the International Centre for Theoretical Sciences. Explore cutting-edge research in Teichmüller theory, focusing on the metric aspects and recent breakthroughs in the Thurston metric. This one-hour lecture contributes to a two-week thematic program that bridges complex analysis, hyperbolic geometry, dynamics, partial differential equations, projective geometry, and other mathematical fields. The program aims to highlight the richness of Teichmüller theory through multiple perspectives and foster collaboration among researchers, including students and postdocs interested in initiating research projects in these emerging areas.