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Teichmüller space is the collection of homotopy-marked Riemann surfaces of a fixed topological type. It has been studied from many perspectives: Teichmüller introduced the space with its eponymous metric measuring conformal distortion; and another well-studied metric is the Weil-Peterson metric. In this talk, we will introduce Thurston's "stretch 'metric'" on Teichmüller space, which is a measure of the minimal global Lipschitz constant needed to send one Riemann surface to another. In a loose analogy with Teichmüller's original work, Thurston constructed a homeomorphism that realizes this constant. The main goal of this talk is a sketch of this construction. Time permitting, we may discuss further work on the theory.