  In this paper we study the Cauchy problem for the elliptic and non-elliptic
derivative nonlinear Schr\"odinger equations in higher spatial dimensions
($n\geq 2$) and some global well-posedness results with small initial data in
critical Besov spaces $B^s_{2,1}$ are obtained. As by-products, the scattering
results with small initial data are also obtained.
