Curved beams are basic structural components of Nano-Electro-Mechanical-Sistems (NEMS) whose design requires appropriate modelling of scale effects. In the present paper, the size-dependent static behaviour of curved elastic nano-beams is investigated by stress-driven nonlocal continuum mechanics. Axial strain and flexural curvature fields are integral convolutions between equilibrated axial force and bending moment fields and an averaging kernel. The nonlocal integral methodology formulated here is the generalization to curved structures of the treatment in [Int. J. Eng. Science 115 (2017) 14-27] confined to straight beams. The corresponding nonlocal differential problem, supplemented with non-standard boundary conditions, is highlighted and shown to lead to mathematically well-posed problems of nano-engineering. The theoretical predictions, exhibiting stiffening nonlocal behaviours, are therefore appropriate to significantly model a wide range of small-scale devices of nanotechnological interest. The nonlocal approach is exploited by analytically establishing size-dependent responses of curved elastic nano-sensors and nano-actuators that are driven by the small-scale characteristic parameter.