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The cubes functor is a direction-symmetric functor from n-categories to n-uple categories. In joint work in progress with Shai Keidar, we prove that the cubes functor is fully faithful and identify its essential image, extending recent results in the case n=2. We also consider the non-univalent setting. There, the cubes functor is not fully faithful, but it is both monadic and comonadic, allowing us to describe non-univalent n-categories in terms of non-univalent n-uple categories equipped with additional structure. These results lead to a simple formula for the Gray tensor product. As an application, we outline the construction of a cubical model for the cobordism categories, and explain how it endows them with a Gray-multiplicative structure.