Inside any 2-category there is a class of maps known as 'normal isofibrations': a generalisation of isofibrations between categories for which isomorphisms can be lifted in a particularly nice way. Bourke and Lack showed that the normal isofibrations equip a nice 2-category with the structure of an ∞-cosmos, the theory of which is due to Riehl and Verity. In this talk, we will generalise the notion of a normal isofibration to the setting of quasicategorically enriched categories. We then show that these normal isofibrations equip a quasicategorically enriched category (so long as it satisfies some minimal axioms) with the structure of an ∞-cosmos. The ∞-cosmoi of quasicategories, complete Segal spaces, Segal categories and 1-complicial sets all turn out to be particularisations of this general phenomenon. This talk is based on joint work with Steve Lack.