This lecture explores Thurston's best Lipschitz maps in the Euclidean case, presented by Athanase Papadopoulos as part of the "New Trends in Teichmüller Theory" program at the International Centre for Theoretical Sciences. Delve into this specialized mathematical topic that represents a recent breakthrough in the Thurston metric theory. The presentation is part of a two-week thematic program focusing on recent developments in Teichmüller theory, which includes areas such as metric theory of Teichmüller spaces, higher Teichmüller theory, Teichmüller theory of infinite-type surfaces, and relations with anti-de Sitter geometry. Learn from renowned experts in a program designed to highlight connections between different mathematical perspectives and encourage collaboration among researchers, including students and postdocs interested in these emerging fields.