The ice sheet–ice shelf transition zone plays an important role in controlling marine ice sheet dynamics, as it determines the rate at which ice flows out of the grounded part of the ice sheet. Together with accumulation, this outflow is the main control on the mass balance of the grounded sheet. In this paper, we verify the results of a boundary layer theory for ice flux in the transition zone against numerical solutions that are able to resolve the transition zone. Very close agreement is obtained, and grid refinement in the transition zone is identified as a critical component in obtaining reliable numerical results. The boundary layer theory confirms that ice flux through the grounding line in a two-dimensional sheet-shelf system increases sharply with ice thickness at the grounding line. This result is then applied to the large-scale dynamics of a marine ice sheet. Our principal results are that (1) marine ice sheets do not exhibit neutral equilibrium but have well-defined, discrete equilibrium profiles; (2) steady grounding lines cannot be stable on reverse bed slopes; and (3) marine ice sheets with overdeepened beds can undergo hysteresis under variations in sea level, accumulation rate, basal slipperiness, and ice viscosity. This hysteretic behavior can in principle explain the retreat of the West Antarctic ice sheet following the Last Glacial Maximum and may play a role in the dynamics of Heinrich events.