A ship's stability is fundamental to the safety of its crew, its cargo, and the environment. Several ocean-going vessels are lost due to instability each year, particularly in high seas. To prevent such losses, a better understanding of ship stability is necessary. In this paper we analyse the stability of ships using advanced mathematical models and methods. All the rigid-body motions of a ship, as well as memory effects in the fluid, are accounted for. The analysis shows that a ship's dynamics depend strongly on the nonlinearities of the ship–fluid system. In our analysis of a particular ship, we notice a sequence of bifurcations when wave heights increase, and we believe that this is an explanation for capsizing. Critical wave heights for capsize were identified. In quartering seas, the required wave height was much lower compared with following seas. A path-following method to determine the stability limits in a systematic manner is being developed.