Let $G$ be a graph with $n$ vertices, and $P$ be a set of $n$ points in the Euclidean space $\R^{m}$. A straight-line embedding of $G$ onto $P$ is an embedding of $G$ onto $P$ whose images of vertices are distinct points in $P$, and images of edges are straight line segments in $\R^{m}$. In this paper, we classify these kinds of sets.