Grounding line migration is a key process affecting the stability of marine ice sheets such as the West Antarctic ice sheet (WAIS). Grounding line motion is often included in numerical models simulating the past and future evolution of the WAIS; however, little attention has been paid to the numerical consistency of these models. The aim of this paper is to assess the ability of simple versions of existing marine ice sheet models to simulate grounding line migration. In particular, we investigate the response of the grounding line to external forcing and the sensitivity of the models' predictions to their numerics and the mechanical coupling between ice sheet and shelf. From the model comparison, there is no consensus on how the grounding line should react to changes in boundary conditions. A crucial finding is the strong dependency of models using a fixed grid on numerical details such as the horizontal grid size. This implies that we should be very wary about grounding line predictions from such models. Including mechanical coupling at the grounding line does not seem to change the qualitative behavior of the models. This suggests that the way the grounding line is treated in marine ice sheet models dominantly determines the grounding line dynamics. We find that models that employ a moving grid to explicitly track the grounding line do not share many of the deficiencies of the fixed grid models. We conclude that at present, no reliable model of the grounding line is available, and further model development is urgently needed.