It is proved that the Cauchy problem for the Kadomtsev-Petviashvili II (KPII) equation is locally well posed in the anisotropic Sobolev spaces H s 1 s 2 (R 2) with s 1 > −⅓ and s 2 ≥ 0, and globally well posed in H s 10(R 2) with s 1 > −.