Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We consider the Cauchy problem associated with the one-dimensional nonlocal derivative nonlinear Schrodinger equation, and establish local well-posedness for 'small' initial data in the usual L(2)-based Sobolev spaces H(s) (R), s > 1/2. We also prove that our result is 'almost sharp' in the sense that the flow-map data-solution fails to be C(3) at the origin from H(s) (R) to H(s) (R) for any s < 1/2. Finally, thanks to the lack of energy conservation, we prove the nonexistence of solitary-wave solutions.916380Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)CNPq [151652/2007-4, 152234/2007-1]