On the well-posedness problem for the derivativenonlinear Schrödinger equation We consider the derivative nonlinear Schrödinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and L^2-critical with respect to scaling. We first discuss whether ensembles of orbits with L^2-equicontinuous initial data remain equicontinuous under evolution. We prove that this is true under the restriction M(q)=∫∣∣q∣∣2<4π. We conjecture that this restriction is unnecessary. Further, we prove that the problem is globally well posed for initial data in H1∕6 under the same restriction on M. Moreover, we show that this restriction would be removed by a successful resolution of our equicontinuity conjecture. 16 5 1245-1270 1245-1270 Mathematical Sciences Publishers