Many densely packed suspensions and colloids exhibit a behavior known as Discontinuous Shear Thickening in which the shear stress jumps dramatically and reversibly as the shear rate is increased. We performed rheometry and video microscopy measurements on a variety of suspensions to determine the mechanism for this behavior. We distinguish Discontinuous Shear Thickening from inertial effects by showing that the latter are characterized by a Reynolds number but are only found for lower packing fractions and higher shear rates than the former. Shear profiles and normal stress measurements indicate that, in the shear thickening regime, stresses are transmitted through frictional rather than viscous interactions. We come to the surprising conclusion that for concentrated suspensions such as cornstarch in water which exhibit the phenomenon of Discontinuous Shear Thickening, the local constitutive relation between stress and shear rate is not necessarily shear thickening. If the suspended particles are heavy enough to settle, we find the onset stress of shear thickening tau(min) corresponds to a hydrostatic pressure from the weight of the particle packing where neighboring particles begin to shear relative to each other. Above tau(min), dilation is seen to cause particles to penetrate the liquid-air interface of the sheared sample. The upper stress boundary tau(max) of the shear thickening regime is shown to roughly match the ratio of surface tension divided by a radius of curvature on the order of the particle size. These results suggest a new model in which the increased dissipation in the shear thickening regime comes from frictional stresses that emerge as dilation is frustrated by a confining stress from surface tension at the liquid-air interface. We generalize this shear thickening mechanism to other sources of a confining stress by showing that, when instead the suspensions are confined by solid walls and have no liquid-air interface, tau(max) is set by the stiffness of the most compliant boundary which frustrates dilation. All of this rheology can be described by a nonlocal constitutive relation in which the local relation between stress and shear rate is shear thinning, but where the stress increase comes from a normal stress term which depends on the global dilation. (C) 2012 The Society of Rheology. [http://dx.doi.org/10.1122/1.4709423]