Answering a question of S. Friedman, Hyttinen and Kulikov, we show that there is a tight connection between the depth of a classifiable shallow theory $T$ and the Borel rank of the isomorphism relation $\cong^\kappa_T$ on its models of size $\kappa$, for $\kappa$ any cardinal satisfying $\kappa^{< \kappa} = \kappa > 2^{\aleph_0}$. This yields a descriptive set-theoretical analogue of Shelah?s Main Gap Theorem. We also discuss some limitations to the possible (Borel) complexities of $\cong^\kappa_T$, and provide a characterization of categoricity of $T$ in terms of the descriptive set-theoretical complexity of $\cong^\kappa_T$. Joint work with F. Mangraviti.