Abstract: 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 fundamental 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.