The purpose of this paper is to survey the new advances in the research on the metric geometry of infinne-dimenstonal Tetchmuller spaces in recent years. It contains the following problems and their solutions: the non-umqueness of geodesic segments; the relation between the uniqueness of segments and the uniqueness of extremal Bel-trarni differentials; non-convexity of spheres and non-differentiability of the Teichmuller metric; isometrically embed-ded polydisks; Busernann points and Strebel points, and their equivalence.
