Abstract: The classification of Zp x Zp actions on the four-sphere plays a central role in the more general study of finite group actions on simply-connected four-manifolds. Orthogonal actions are easy to understand, and form our starting point. We discuss the extent to which general (locally linear) actions resemble linear actions, first from the point of view of homological fixed-point theory, then with some simple constructions of nonlinear examples, and finally with an application of surgery theory to classify these actions up to an appropriate notion of concordance. The only obstruction to concordance turns out to be a codimension two Arf invariant. We discuss its geometrical meaning, and briefly discuss the still-open question of its full realizability.