  In this paper, we study a notion of what we call vertex Leibniz algebra. This
notion naturally extends that of vertex algebra without vacuum, which was
previously introduced by Huang and Lepowsky. We show that every vertex algebra
without vacuum can be naturally extended to a vertex algebra. On the other
hand, we show that a vertex Leibniz algebra can be embedded into a vertex
algebra if and only if it admits a faithful module. To each vertex Leibniz
algebra we associate a vertex algebra without vacuum which is universal to the
forgetful functor. Furthermore, from any Leibniz algebra $\g$ we construct a
vertex Leibniz algebra $V_{\g}$ and show that $V_{\g}$ can be embedded into a
vertex algebra if and only if $\g$ is a Lie algebra.
