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Eukaryote cells have flexible membranes that allow them to have a variety of dynamical shapes. The shapes of the cells serve important biological functions, both for cells within an intact tissue, and during embryogenesis and cellular motility. How cells control their shapes and the structures that they form on their surface has been a subject of intensive biological research, exposing the building blocks that cells use to deform their membranes. These processes have also drawn the interest of theoretical physicists, aiming to develop models based on physics, chemistry and nonlinear dynamics. Such models explore quantitatively different possible mechanisms that the cells can employ to initiate the spontaneous formation of shapes and patterns on their membranes. We review here theoretical work where one such class of mechanisms was investigated: the coupling between curved membrane proteins, and the cytoskeletal forces that they recruit. Theory indicates that this coupling gives rise to a rich variety of membrane shapes and dynamics, while experiments indicate that this mechanism appears to drive many cellular shape changes. This article is part of the theme issue ‘Self-organization in cell biology’.

Guided by curvature: shaping cells by coupling curved membrane proteins and cytoskeletal forces