<p>Topological superconductivity can be induced in an s-wave superconductor by an adjacent magnetic layer with a non-collinear spin structure. Good candidates are atomic-scale spin textures with large tilting angles between neighboring spins which exhibit intriguing properties such as topological orbital moments and topological Hall conductivity. Here we investigate the coupling of such non-coplanar spin structures to an adjacent superconducting layer based on a tight-binding model. We consider spin structures recently observed in ultrathin Fe and Mn-based films on the Re(0001) surface such as the triple-Q state, atomic- and nano-scale skyrmion lattices, and study the effect of a continuous distortion of the spin state. We characterize the topology of the spin texture via the scalar spin chirality and the topology of the superconductor by its Chern number. We find that a non-zero total scalar spin chirality, leads to a gapped topological superconducting phase while only trivial superconducting phases can appear for a vanishing value. Our study shows that the size of the superconducting gap in the topological phase rises with the total scalar spin chirality. For an atomic-scale skyrmion lattice, we predict the existence of chiral edge modes on a realistic island structure detectable by the local density of states or supercurrents.</p>