In this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces $\dot {H}^{s}_{p}(\mathbb {R}^{d})$ for $d \geq 2, p > \frac {d}{2}$, and $\frac {d}{p} - 1 \leq s < \frac {d}{2p}$. The obtained result improves the known ones for $p > d$ and $s = 0$ (see [4, 6]). In the case of critical indexes $s=\frac {d}{p}-1$, we prove global well-posedness for Navier-Stokes equations when the norm of the initial value is small enough. This result is a generalization of the one in [5] in which $p = d$ and $s = 0$.