The mechanics of ship capsize under steady and transient conditions is reviewed, focusing on recent applications of global geometrical techniques of nonlinear dynamics. These yield significant new ideas about capsize in waves and its generalization, the escape of a driven oscillator from a potential well. These ideas are robust against gross changes in the forms of the stiffness and damping functions. Fractal basin boundaries in phase and control space yield useful design criteria against transient capsize, which have been applied to real ships. Invariant manifolds are used to explain and predict the sudden loss of safe basin in the space of the starting conditions, and indeterminate resonant jumps to capsize. Further work is concerned with capsize suppression by heave-roll coupling; effects of parametric excitation; and capsize under a propagating wave front. After this historical review, the practical relevance of the results is assessed, and suggestions are made for a standardized transient testing procedure for hulls. A systematic formulation for rolling in beam waves, employing the effective gravitational field perpendicular to the wave surface and the Froude-Krilov assumption, allows the use of the calm-water GZ curve. With general stiffness and damping functions, dimensional analysis offers insights that are often overlooked: for example, the sustainable wave slope is always proportional to the angle of vanishing stability. A degree of quantification is provided by a design formula derived from the displacement magnification of linear resonance. This is validated by Melnikov theory and simulation. It predicts that under worst-case excitation we have: sustainable wave slope = 2 ζθv , where θv is the angle of vanishing stability and ζ is a damping ratio appropriate for heavy roll. So in ocean waves of slope 0.5 (≈30°), a vessel with a θv of one radian needs a damping ratio of about 1/4. Implications for the design of hulls reveal counter-intuitive results: it is the distance of the potential barrier, not its height, that prevents escape or capsize. The formula helps to define a universal capsize diagram. New results on symmetry breaking are finally presented. These show that capsize studies of a symmetric unbiased vessel can give seriously unsafe results. The sustainable wave slope is so sensitive to a symmetry-breaking bias (due to wind or cargo imbalance) that a static heel of 2.5° can halve the sustainable slope over a wide range of sea states. This review article has 112 references.