In extremely thin ferromagnetic films, an additional interaction, the so-called Dzyaloshinskii-Moriya interaction
(DMI), arises in the micromagnetic energy along with a magnetocrystalline anisotropy favoring the
magnetization to be out-of-plane. In such materials, topoligically nontrivial, point-like configurations of the
magnetization called magnetic skyrmions are observed, which are of great interest in the physics community
due to possible applications in high-density data storage. We characterize skyrmions as, typically, local
minimizers of a micromagnetic energy augmented by DMI in a variety of settings. As no degree ±1 harmonic map
from the plane to the sphere has square-integrable out-of-plane components, quantitative rigidity estimates for
such maps are required to capture the asymptotics in the regime of dominating exchange (or Dirichlet) energy.
We will furthermore discuss recent advances towards an understanding of multiple skyrmions.