Landau-Lifshitz equations are the basic dynamical equations in a micromagnetic description of a ferromagnet. They are naturally viewed as geometric evolution PDE of dispersive "Schrödinger map" or mixed dispersive-diffusive type, which scale critically with respect to the physical energy in two dimensions. We review some recent results and open questions concerning the interrelated questions of stability of topological solitons, asymptotic behaviour, and (non-)formation of singularities.