We study the classification problem of Polish metric spaces up to isometry and the isometry groups of Polish metric spaces. In the framework of the descriptive set theory of definable equivalence relations, we determine the exact complexity of various classification problems concerning Polish metric spaces. We start with the class of all Polish metric spaces and prove that it is Borel bireducible to the universal orbit equivalence relation induced by Borel actions of Polish groups. We then turn to special classes of Polish metric spaces, including locally compact, ultrametric, zero-dimensional, homogeneous, and ultrahomogeneous spaces. In the investigation of the classification problems we also obtain characterizations for isometry groups of various classes of Polish metric spaces.
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