The theoretical determinants of maximum sustainable government debt are investigated using Diamond’s overlapping-generations model. A level of debt is defined to be ‘sustainable’ if a steady state with non-degenerate values of economic variables exists. We show that a maximum sustainable level of debt almost always exists. Most interestingly, it normally occurs at a ‘catastrophe’ rather than a ‘degeneracy’, i.e. where variables such as capital and consumption are in the interiors, rather than at the limits, of their economically meaningful ranges. This means that if debt is increased step by step, the economy may suddenly collapse without obvious warning.