In the 1960s Atiyah and Kodaira constructed surface bundles over surfaces with many interesting properties. The topology of such a bundle is completely encoded by its monodromy representation (a homomorphism to a mapping class group), and it is a basic problem to understand precisely how the topology of the bundle is reflected in algebraic properties of the monodromy. The main result of this talk is that the Atiyah–Kodaira bundles have arithmetic monodromy groups. A corollary of this result is that Atiyah–Kodaira bundles fiber in exactly two ways. This is joint work with Nick Salter.