  We introduce the notion of vertex coalgebra, a generalization of vertex
operator coalgebras. Next we investigate forms of cocommutativity,
coassociativity, skew-symmetry, and an endomorphism, $D^*$, which hold on
vertex coalgebras. The former two properties require grading. We then discuss
comodule structure. We conclude by discussing instances where graded vertex
coalgebras appear, particularly as related to Primc's vertex Lie algebra and
(universal) enveloping vertex algebras.
