Summary: Stokes (1845) developed a general constitutive relation which admitted the possibility that the viscosity could depend on the pressure. Such an assumption is particularly well suited to modelling flows of many fluids at high pressures, and is relevant to several flow situations involving lubricants. Fluid models in which the viscosity depends on pressure have not received the attention, and we consider unidirectional and two-dimensional flows of such fluids here. We note that solutions can have markedly different characteristics than the corresponding solutions for classical Navier-Stokes fluid. It is shown that unidirectional flows corresponding to Couette or Poiseuille flow are possible only for special forms of viscosity. Furthermore, we show that interesting non-unique solutions are possible for flow between moving plates, which has no counterpart in the classical Navier-Stokes theory. We also study, numerically, two-dimensional flows that are technologically significant: flows between rotating coaxial or eccentric cylinders, and a flow across a slot. The solutions provide interesting departures from those for the classical Navier-Stokes fluid.