From experimental results obtained by Crook (1963) and by Smith (1960) from the field of elastohydrodynamic lubrication, it must be concluded that in certain circumstances mineral oils exhibit non-Newtonian behaviour, i.e. the shear stress is no longer proportional to the shear rate, at constant temperature. In this paper, we seek to describe the relations between shear stress and shear rate, which have been reported for mineral oils under various experimental conditions, and to relate them to properties of mineral oils reported from other fields. We start with the supposition that the non-Newtonian behaviour is caused by the viscoelastic properties of the oil. From the theory given by Oldroyd (1958) of the behaviour of a viscoelastic substance in continuous shear at high shear rates, it is concluded that, under certain simplifying conditions, there should be a simple relation between the stress system in continuous shear at a shear rate and the components of the complex shear modulus G* in oscillatory shear at an angular frequency (o, with . Here K is an unknown shift constant. This conclusion is confirmed within a certain range of conditions by an examination of experimental results, particularly those of Russell (1946), obtained with substances showing viscoelastic behaviour under conditions more easily reproducible under controlled laboratory conditions than those required for mineral oils. We then consider the results for mineral oils obtained by Crook (1963) and by Smith (1960) under conditions of elastohydrodynamic lubrication. These are fitted to the results obtained in oscillatory shear by Barlow & Lamb (1959) and satisfactory agreement is obtained. Whereas for other viscoelastic materials K~ 1, apparently independent of pressure, we find that for mineral oils if decreases with increasing pressure and increases with increasing temperature. Extrapolation to atmospheric pressure gives K ~ 103, which is consistent with the results obtained by Porter & Johnson (1959), who used a rotational viscometer. A tentative interpretation of the variation of K in terms of that of the Oldroyd parameter is given. Certain features of the experimental evidence so far available are unsatisfactory, and more work is needed.