Experiments beneath Briedamerkurj¿kull in Iceland have led to development of flow laws for the subglacial till, relating strain rate to shear stress and effective pressure and assuming either Bingham fluid or nonlinearly viscous fluid behavior. Water pressures in the till are less than ice pressures and it is suggested that this may lead to infilterationof ice into the sediment, which inhibits sliding at the ice/sediment interface. Where water pressures are equal or near to ice pressures, infilteration does not occur and sliding may result. A one-dimensional theory of subglacial deformation is developed in which the empirical flow law is coupled with a model of subglacial hydrology and consolidation. This predicts stable states in which subglacial sediment either does not deform or a dilatant deforming horizon forms with positive effective pressures at the ice/bed interface or unstable states where zero or negative effective pressures are predicted. Time dependent analyses show that response times following perturbations of the system may be of the order of 103 years and thus that unsteady behavior may be normal on glaciers flowing over unlithified sediment beds. It is suggested that the natural variability of material properties in subglacial sediment beds leads to the development of drumlins on the glacier bed. It is suggested that unstable deformation at zero or negative effective stress leads to ''piping'' in subglacial sediments at the glacier terminus and the growht of sediment-floored, subglacial tunnels. Their frequency is that which is sufficient to draw down subglacial water pressres so as to prevent unstable deformation. Where they discharge large water volumes, subglacial sediments flow laterally toward them producing ''tunnel valleys.'' This sediment is then removed by water flowing along the axial tunnel. Tunnel valleys can be regarded as the equivalent in soft sediment areas of eskers in bedrock areas. ¿ American Geophysical Union 1987