This paper discusses the mechanism of rolling friction under conditions where the deformations involved are predominantly elastic. Experiments on the rolling of a metal cylinder over a rubber surface show that interfacial slip of the type described by Reynolds is minute and totally insufficient to account for the observed resistance to rolling. It is shown quantitatively that the rolling resistance under these conditions is due to elastic hysteresis losses in the rubber. This accounts for the ineffectiveness of lubricants in reducing the rolling friction. Similar results are obtained for hard spheres rolling on rubber surfaces. If, however, a sphere is rolled in a performed rubber groove, interfacial slip between the ball and groove may occur since the central band on the ball measures out a larger circle than the bands at the edge of the groove. A simple quantitative theory of this effect is given. This type of slip, first described by Heathcote, is very marked when the groove is deep and of curvature very close to that of the ball; under these conditions a suitable lubricant can effect a considerable reduction in the rolling resistance. For shallower grooves the differential slip is reduced and the Heathcote contribution to the observed rolling resistance becomes trivial. These conclusions are applied to the rolling friction of a hard steel sphere in the equilibrium groove formed in the surface of a softer metal (part I). If the rolling friction is attributed to the Heathcote mechanism a coefficient of friction within the ellipse of contact of the order of $\mu $ = 1 to 2 must be invoked even in the presence of the best boundary lubricants. This is impossibly high. If, on the other hand, the rolling friction is attributed to hysteresis losses, large loss factors, greater than 20%, must be invoked for example for copper surfaces. In order to resolve this difficulty a new experimental approach has been adopted in which a copper ball is rolled over an identical copper ball. Because of symmetry conditions at the region of contact both the Reynolds and the Heathcote type of slip are eliminated. The equilibrium rolling friction is found to be as high as that observed when a hard steel sphere rolls in a copper groove. It is concluded that interfacial slip contributes little to the rolling friction although it may play an important part in surface wear. Consequently, lubricants may reduce the amount of wear but have little effect on the rolling resistance. The greater part of this arises from elastic hysteresis losses within the metals themselves.