On the classification of Polish metric spaces up to isometry

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