Curvature and geometry have significant implications in fundamental physics, leading to the appearance of intriguing novel phenomena. In the field of nanomagnetism, geometrical-induced effects yield important consequences, which, despite their relevance for domain wall (DW) motion-based applications, still await experimental validation. In this letter, a spiral-shaped magnetic nanostructure is used to demonstrate experimentally that curvature gradients determine DW motion. A saturating magnetic field is applied to the spirals to induce the magnetic onion state, generating head-to-head (HtH) and tail-to-tail (TtT) DW. Curvature gradient promotes domain wall motion through a local curvature-dependent effective force, toward regions of higher curvature. These effects have been studied by measuring depinning fields and supported by micromagnetic simulations and an analytical model. Our results show the potential of curvature engineering for the realization of low-energy spintronic devices.