The advance of a glacier over a deforming sediment layer is analysed numerically. We treat this problem as a contact problem involving two slowly-deforming viscous bodies. The surface evolution of the two bodies, and of the contact interface between them, is followed through time. Using various different non-linear till rheologies, we show how the mode of advance depends on the relative effective viscosities of ice and till. Three modes of advances are observed: (1) overriding, where the glacier advances through ice deformation only and without deforming the sediment; (2) plug-flow, where the sediment is strongly deformed, the ice moves forward as a block and a bulge is built in front of the glacier; and (3) mixed-flow, where the glacier advances through both ice and sediment deformation. For the cases of both overriding and mixed-flow, an inverse depth-age relationship within the ice is obtained. A series of model experiments show the contrast in effective viscosity between ice and till to be the single most important model parameter defining the mode of advance and the resulting thickness distribution of the till. Our model experiments indicate that the thickness of the deforming till layer is greatest close to the glacier front. Measurements of till thickness taken in such locations may not be representative of deforming till thickness elsewhere. Given sufficiently large contrast in effective viscosity between ice and till, a sediment bulge is formed in front of the glacier. During glacier advance, the bulge quickly reaches a steady state form strongly resembling single-crested push moraines. Inspection of particle paths within the sediment bulge, shows that particles within the till travel at a different speed from the bulge itself, and the push moraine to advance as a form-conserving non-linear wave.