  We study the Cauchy problem for the Klein-Gordon-Zakharov system in spatial
dimension $d \ge 4$ with radial or non-radial initial datum $(u, \partial_t u,
n, \partial_t n)|_{t=0}\in H^{s+1}(\mathbb{R}^d) \times H^s(\mathbb{R}^d)
\times \dot{H}^s(\mathbb{R}^d) \times \dot{H}^{s-1}(\mathbb{R}^d)$. The
critical value of $s$ is $s=s_c=d/2-2$. If the initial datum is radial, then we
prove the small data global well-posedness and scattering at the critical space
in $d \ge 4$ by applying the radial Strichartz estimates and $U^2, V^2$ type
spaces. On the other hand, if the initial datum is non-radial, then we prove
the local well-posedness at $s=1/4$ when $d=4$ and $s=s_c+1/(d+1)$ when $d \ge
5$ by applying the $U^2, V^2$ type spaces.
