A Walker $n$-manifold is a pseudo-Riemannian manifold which admits a field of parallel null $r$-planes, with $r\leq \frac{n}{2}$. The canonical forms of the metrics were investigated by A. G. Walker \cite{walker}. Of special interest are the even-dimensional Walker manifolds $(n=2m)$ with fields of parallel null planes of half dimension $(r=m)$. In this paper, we investigate geometric properties of some curvature tensors of an eight-dimensional Walker manifold. Theorems for the metric to be Einstein, locally conformally flat and for the Walker eight-manifold to admit a Kähler structure are given.