  We consider the generalized two-dimensional Zakharov-Kuznetsov equation
$u_t+\partial_x \Delta u+\partial_x(u^{k+1})=0$, where $k\geq3$ is an integer
number. For $k\geq8$ we prove local well-posedness in the $L^2$-based Sobolev
spaces $H^s(\mathbb{R}^2)$, where $s$ is greater than the critical scaling
index $s_k=1-2/k$. For $k\geq 3$ we also establish a sharp criteria to obtain
global $H^1(\R^2)$ solutions. A nonlinear scattering result in $H^1(\R^2)$ is
also established assuming the initial data is small and belongs to a suitable
Lebesgue space.
