The focus of the present theme issue is on significant recent advances in the understanding and modelling of floating sea ice. This also follows an extended and concerted research programme held in late 2017 at the Isaac Newton Institute in which the mathematics of sea-ice phenomena was addressed. Other forms of ice are included where they help with the phenomenological and/or methodological understanding of sea-ice behaviour. The heart of the theme here is associated with consistent modelling of facets of sea-ice mechanics and thermodynamics, as well as sea-ice interactions with fluids and solids. Particular studies are on multi-scale modelling of ice characteristics and behaviour, ice–fluid interaction, coupled problems of ice-structure interaction, and ice fracture and cracks. Environmental aspects of ice-related problems are included when they relate to sea-ice mechanics, and congruent industrial icing problems are also discussed. The topic is appropriate for a theme issue because of its depth, breadth and timeliness. The content is of a fully inter-disciplinary nature throughout. The new developments presented and discussed hinge on the advances of modelling of ice-related problems giving an appropriate level of both physical and mathematical rigour to such problems. The theme aims to help towards defining a state of the art in floating ice research, but especially …