Mammalian cells adhere to the extracellular matrix (ECM) and sense mechanical cues through integrin-mediated adhesions 1, 2 . Focal adhesions and related structures are the primary architectures that transmit forces between the ECM and the actin cytoskeleton. Although focal adhesions are abundant when cells are cultured on rigid substrates, they are sparse in soft environments that cannot support high mechanical tensions 3 . Here, we report a new class of integrin-mediated adhesions, curved adhesions, whose formation is regulated by membrane curvature instead of mechanical tension. In soft matrices made of protein fibres, curved adhesions are induced by membrane curvatures imposed by the fibre geometry. Curved adhesions are mediated by integrin ɑVβ5 and are molecularly distinct from focal adhesions and clathrin lattices. The molecular mechanism involves a previously unknown interaction between integrin β5 and a curvature-sensing protein FCHo2. We find that curved adhesions are prevalent in physiologically relevant environments. Disruption of curved adhesions by knocking down integrin β5 or FCHo2 abolishes the migration of multiple cancer cell lines in 3D matrices. These findings provide a mechanism of cell anchorage to natural protein fibres that are too soft to support the formation of focal adhesions. Given their functional importance for 3D cell migration, curved adhesions may serve as a therapeutic target for future development.