In this talk,we shall discuss the structure of cocommutative vertex bialgebras. For a general vertex bialgebra V, we show that the set G(V) of group-like elements is naturally an abelian semigroup, whereas the set P(V) of primitive elements is a vertex Lie algebra. For $g\in G(V)$, denote by $V_g$ the connected component containing g. Among the main results, we show that if V is a cocommutative vertex bialgebra, then $V=\oplus_{g\in G(V)}V_g$, where $V_1$ is a vertex subbialgebra which is isomorphic to the vertex bialgebra $\mathcal{V}_{P(V)}$ associated to the vertex Lie algebra P(V), and $V_g$ is a $V_1$-module for $g\in G(V)$. In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to $\mathcal{V}_{P(V)}$ and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. Furthermore, under the condition that G(V) is a group and lies in the center of V, we prove that $V=\mathcal{V}_{P(V)}\otimes C[G(V)]$ as a coalgebra where the vertex algebra structure is explicitly determined. This talk is based on a joint work with Jianzhi Han and Yukun Xiao.