A fundamental understanding of magnetic phenomena requires the determination of spin structures down to the atomic scale. The direct visualization of atomic-scale spin structures has been accomplished by combining the atomic resolution capability of Scanning Tunnelling Microscopy (STM) with spin sensitivity, based on vacuum tunnelling of spin-polarized electrons. The resulting technique, Spin-Polarized Scanning Tunnelling Microscopy (SP-STM), nowadays provides unprecedented insight into collinear and non-collinear spin structures at surfaces of magnetic nanostructures and has already led to the discovery of new types of magnetic order at the nanometre scale. Several examples of complex magnetic order as revealed by atomic-resolution SP-STM will be reviewed.