  The Zakharov-Kuznetsov equation in space dimension $d\geq 3$ is considered.
It is proved that the Cauchy problem is locally well-posed in
$H^s(\mathbb{R}^d)$ in the full subcritical range $s>(d-4)/2$, which is optimal
up to the endpoint. As a corollary, global well-posedness in
$L^2(\mathbb{R}^3)$ and, under a smallness condition, in $H^1(\mathbb{R}^4)$,
follow.
