To understand how viscous and elastic membrane forces mediate the adhesion of fluid-borne cells to biological surfaces under the action of specific receptor–ligand bonds, we consider a model problem in which a two-dimensional cell interacts with a plane adhesive surface. The cell is modelled as an extensible membrane under tension containing fluid of constant volume. Assuming rapid binding kinetics, molecular binding forces are described through a contact potential that is long-range attractive but short-range repulsive. Using lubrication theory to describe the thin-film flow between the cell and the plane, we model sedimentation of the cell onto the plane under adhesive forces, followed by removal of the cell from the plane under the action of an external force. Numerical simulations show how these events are dominated respectively by quasi-steady spreading and peeling motions, which we capture using an asymptotic analysis. The analysis is extended to model a cell tank-treading over an adhesive wall in an external shear flow. The relation between cell rolling speed and shear rate is determined: at low speeds it is linear and independent of the viscosity of the suspending fluid; at higher speeds it is nonlinear and viscosity-dependent.