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Quantification of the risk posed by submarine landslides is an essential part of the design process for offshore oil and gas developments and also for prediction and mitigation of potentially devastating tsunamis. A number of studies have been conducted to quantify the initiation of catastrophic failure in planar slopes triggered by undercutting at the toe or the formation of a thin lens of weakened material within the slope due, for example, to seismic activity or hydrate dissociation. However, there has been more limited investigation of instability problems of curvilinear slopes, even though these are more typical of natural slopes. The present paper explores the propagation of a shear band initiated from the centre, that is, the steepest point, of generic curvilinear slopes. The study extends the process zone method to establish governing equations for curvilinear shear band propagation with different geometry functions, which are then solved analytically and numerically to derive the shear band length. Criteria for stable propagation of the shear band (‘progressive failure’) and unstable propagation (‘catastrophic failure’) are formulated in terms of the maximum gravitational shear stress ratio and the shape of the slope. A simple method is proposed for practical application to quantify the status of submarine slopes, identifying the transition to catastrophic failure. The findings are compared with existing criteria and discussed with respect to the effects of the soil inertia in the early stages of shear band development.