We call magnetization swirls small vortex-like micromagnetic structures at the surface of soft magnetic (low-anisotropy) materials which are magnetized perpendicular to the surface in their core region. In this paper the occurrence of swirls is discussed and their structure is explored by a Ritz method calculation. The main result is that the radius of a swirl should not be larger than about [MATH], amounting to some 10 nm in typical materials. We can observe the characteristic outer vortex pattern of swirls by high resolution Kerr microscopy. Other indirect evidence for the existence of swirls and for their role is reviewed and the possibilities for their direct observation by modern high-resolution methods are discussed