  We prove local and global well-posedness in $H^{s,0}(\mathbb{R}^{2})$, $s >
-1/2$, for the Cauchy problem associated with the
Kadomotsev-Petviashvili-Burgers-I equation (KPBI) by working in Bourgain's type
spaces. This result is almost sharp if one requires the flow-map to be smooth.
