While Polish metric spaces are usually uncountable, they behave like countable models because they are separable. But, how do we study metric spaces within model theory? Either we make the space suit classical model theory: instead of the distance function we have a countable collection (R_q) of binary relations, saying that the distance is less than the rational q. Or we adapt model theory to better suit metric spaces: relations become real valued functions, and in particular equality is now the distance function. This yields the field of continuous model theory. In the setting of classical model theory, S.Friedman, Fokina, Koerwien and Nies (2012) produced Polish metric spaces of arbitrarily high countable Scott rank. Answering their question, a paper by Michal Doucha recently submitted shows that the Scott rank is countable. In the setting of continuous model theory, isometry is too fine, and is better replaced by having Gromov-Hausdorff distance 0. The natural analog of the Scott rank is countable, and the terminal Scott function yields this distance. In recent work Ben Yaacov, Nies and Tsankov were naturally led to versions of the Lopez Escobar and Craig Interpolation theorems in the setting of infinitary continuous logic for separable metric models.