Moduli spaces of rank 2 vector bundles on P^2, after Wolf Barth. In his classical work Wolf Barth invesitigates the geometry of the moduli space M of vector bundles on the projective plane P^2 of Chern class (rk,c1,c2)=(2,0,n). First, given a (stable) vector bundle E Barth constructs a net (linera system of dimension two) of quadrics in the n-dimensional space. Barth proves that this gives an embedding of the moduli space M to the moduli space of nets of quadrics. Second, having a net of quadric, one can classically associate to it a curve S of degree n on P^2 and a line bundle on S which half the canonical class (theta-characteristics). This gives an embedding of M into the moduli spaces of (plane curve of degree n, theta-characteristics on it). Third, Barth proves that the curve S is isomorphic to a curve of jumping lines for E: it is the moduli of lines on P^2 the restriction to which of E is not isomorphic to O+O. Forgetting the theta-characteristics, we get a (finite) morphism from M to the moduli space of degree n plane curves. Barth proves that for n=4 the image of this morphism is the set of so-colled Luroth quartics, and for n = 5 the image is the set of Darboux quintics.