We use the theory of graph limits to study several quasi-random properties, mainly dealing with various versions of hereditary subgraph counts. The main idea is to transfer the properties of (sequences of) graphs to properties of graphons, and to show that the resulting graphon properties only can be satisfied by constant graphons. These quasi-random properties have been studied before by other authors, but our approach gives proofs that we find cleaner, and which avoid the error terms and in the traditional arguments using the Szemerédi regularity lemma. On the other hand, other technical problems sometimes arise in analysing the graphon properties; in particular, a measure-theoretic problem on elimination of null sets that arises in this way is treated in an appendix.
