In joint work with Dominic Verity we prove that four models of (∞,1)-categories — quasi-categories, complete Segal spaces, Segal categories, and 1-complicial sets — are equivalent for the purpose of developing ∞-category theory. To prove this we first introduce the notion of an ∞-cosmos, a category in which ∞-categories live as objects, an example of which is given by each of the four models mentioned above. We then explain how the category theory of ∞-categories can be developed inside any ∞-cosmos; eg, we define right adjoints and limits and prove that the former preserve the latter. We conclude by arguing that the four above mentioned ∞-cosmoi all biequivalent, the upshot being that ∞-categorical structures are preserved, reflected, and created by a number of “change-of-model” functors. More precisely, we show that each of these ∞-cosmoi have a biequivalent “calculus of modules,” modules between ∞-categories being a vehicle to express ∞-categorical universal properties.