Cells can migrate individually between tissues and organs or collectively in tightly or loosely associated groups. In both cases, cell migration is regulated by extracellular cues of very different natures. Here I will describe a cellular mechanism called curvotaxis that enables individual cells and growing epithelial monolayers to sense and respond to curvature variations from the substrate on which they adhere. Live imaging combined with functional analysis and in silico modeling shows that curvotaxis relies on a dynamic interplay between the nucleus and the cytoskeleton - the nucleus acting as a negative curvature magnet that leads the migrating cell towards concave curvatures. Curvature affects also focal adhesions organization and dynamics, nuclear shape and gene expression. Growing epithelial colonies can also respond to curvature by a process involving both curvotaxis in the leading edge where cells are loosely associated, and oriented cell divisions in the more mature parts of the colony. Altogether, this work identifies cell-scale curvature as an essential physical cue that can guide both single and collective cell migration processes. Potential applications in tissue engineering and biomaterial design will be discussed.