In this talk we will present recent results concerning
the initial value problem (IVP) associated to generalized derivative Schrödinger
equations. We show local well-posedness for small initial data in a suitable weighted
Sobolev spaces. We use an argument introduced by Cazenave and Naumkin to
obtain our main results combined with the homogeneus and inhomogeneous smoothing
effects of Kato type. If time permits we will show how these results can be extended for
any data size in a suitable class.