We consider lubrication theory for a two-dimensional viscoplastic fluid confined between rigid moving boundaries. A general formulation is presented which allows the flow field and pressure to be calculated given an arbitrary theological model: the Herschel-Bulkley law is used for illustration. The theory is first applied to a (full) viscoplastic journal bearing with arbitrary motions allowed for the inner cylinder (either prescribed, or arising from an imposed load and torque). Conditions are derived determining when motion is arrested by the yield stress. We next apply the theory to a slider bearing filled with Bingham fluid, computing the lift force on the bearing and the fluid flux through it. The results are then extended to model an inclined plate that is towed at constant horizontal speed over a shallow viscoplastic layer but is able to move vertically. Steady planing solutions are stable at low towing speeds, but give way to unstable vertical oscillations of the plate at higher speed; the yield stress has a relatively weak effect on this instability. The pattern imprinted on the fluid layer by the oscillations provides an analogue of the washboard phenomenon on gravel roads. (C) 2011 Elsevier B.V. All rights reserved.