We will focus our attention on moduli spaces of semi-stable coherent sheaves on the projective plane, of rank 2, first Chern class being 0. Semi-stable for a coherent sheaf means without any torsion, and also a condition on the "slopes" of its coherent subsheaves I will first recall. There are two ways to construct such moduli spaces: first by taking some quotient in the sense of geometric invariant theory of some open subset of some Hilbert scheme, the second consisting in using the description of such semi-stable sheaves as cohomology of monads (Beilinson monads). There is a Barth morphism from such a moduli space to a projective space of plane curves, which sends a semi-stable sheaf to its curve of jumping lines. Le Potier and Tikhomirov have shown 5 years ago that it is generally injective. We use this fact to compute the degree of some subvariety of remarkable plane curves, the "Poncelet curves". I could also explain another part of my PhD work about some computations of dimensions of some linear systems on the moduli space M_4 (rank 2, Chern classes 0,4), and show why it is useful for the question of "strange duality" (roughly, this is the analogue for the plane of Verlinde's formula problem). If there is some time left, I would like to explain my current work, which deals with positivity properties of symmetric twisted powers of a very general semi-stable vector bundle of rank 2, first Chern class 0 on the plane.