Deformations of $G_2$ and $\Spin(7)$ Structures We consider some deformations of $G_2$-structures on $7$-manifolds. We discover a canonical way to deform a $G_2$-structure by a vector field in which the associated metric gets twisted'' in some way by the vector cross product. We present a system of partial differential equations for an unknown vector field $w$ whose solution would yield a manifold with holonomy $G_2$. Similarly we consider analogous constructions for $\Spin(7)$-structures on $8$-manifolds. Some of the results carry over directly, while others do not because of the increased complexity of the $\Spin(7)$ case. Keywords:$G_2 \Spin(7)$, holonomy, metrics, cross productCategories:53C26, 53C29