Rapid removal of small-sized droplets passively using fixed structures is a key challenge for various applications including anti-icing, rapid cooling, and water harvesting. In this work, we investigate the directional motion of nanodroplets on axisymmetric surfaces with curvature gradient through molecular dynamics (MD) simulations. It is found that as the shape of the axisymmetric surface is changed from a dome to a trumpet, the droplet velocity is greatly enhanced, by a factor of ∼14. Such an increase is mainly caused by the increment in the driving force. The droplet velocity changes nonlinearly as the surface wettability is varied and assumes the maximum at the contact angle of ∼75°. We derive a formula for the driving force of nanodroplets on general axisymmetric surfaces by evaluating the pressure gradient inside the droplet induced by the curvature gradient. Molecular dynamics simulations are performed to directly measure the driving force and confirm that the theoretical formula works well. By illustrating the reduced initial velocity of droplets as a function of a dimensionless number, which represents the ratio of the driving force to the retentive force due to contact angle hysteresis, we show that the onset of droplet motion on axisymmetric surfaces occurs when the dimensionless number is above a critical value. The dimensionless number reveals the effects of surface geometry, surface wettability, and droplet size on the droplet motion.