  This article is concerned with the Zakharov-Kuznetsov equation {equation}
\label{ZK0} \partial_tu+\partial_x\Delta u+u\partial_xu=0 . {equation} We prove
that the associated initial value problem is locally well-posed in $H^s(\mathbb
R^2)$ for $s>\frac12$ and globally well-posed in $H^1(\mathbb R\times \mathbb
T)$ and in $H^s(\R^3) $ for $ s>1$. Our main new ingredient is a bilinear
Strichartz estimate in the context of Bourgain's spaces which allows to control
the high-low frequency interactions appearing in the nonlinearity of
\eqref{ZK0}. In the $\mathbb R^2$ case, we also need to use a recent result by
Carbery, Kenig and Ziesler on sharp Strichartz estimates for homogeneous
dispersive operators. Finally, to prove the global well-posedness result in $
\R^3 $, we need to use the atomic spaces introduced by Koch and Tataru.
