Killing spinors and Weierstrass representations of surfaces in 3 and 4 dimensions

In the last two decades a variety of formulae have been proposed to parametrize conformal immersions of surfaces in 3 or 4 dimensions, often named "Weierstrass formula", referring to the classical formula of Weierstrass and Enneper for minimal surfaces. Most of these expressions rely on "spinors" satisfying a "Dirac equation" (Taimanov, Konopelchenko, Kusner, Schmitt etc.). We will explain where these formulae come from and how they are related to more recent work on classical spinors by Bär, Friedrich and Roth. If time allows, I will also mention the point of view of isometric immersions and moving frames.