Abstract:   The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in $H^{s}({\Real^2})$ for all $\frac{5}{7}<s<1$ via using I-method in the context of atomic spaces. By means of the increment of modified energy, the existence of global attractor for the weakly damped, forced Zakharov-Kuznetsov equation is  established in $H^{s}({\Real^2})$ for $\frac{10}{11}<s<1$.