In this joint work with Galina Perelman, we are interested in the issue of global well-posedness for the nonlinear cubic Schrödinger equation with derivative on the real line. This equation called (DNLS) appeared in the eighties in the study of the asymptotic regimes of the propagation of Alfvén waves in magnetized plasmas. The local well-posedness for this equation is well understood in the scale of Sobolev spaces since twenty years ago while the understanding of the global well-posedness is rather recent. In comparison with other integrable equations like the nonlinear cubic Schrödinger equation (NLS), the (DNLS) equation displays a lack of coercivity and new ideas were needed to show that the solutions are global. In this joint work with Galina Perelman, by combining the techniques of profile decompositions with the integrable structure of the equation, we were able to establish a global well-posedness result for (DNLS) in $H^s$, $s \geq 1/2$.