This talk is on the interactions among vertex algebras, vertex Lie algebras, and vertex bialgebras. From definition, every vertex algebra is naturally a conformal Lie algebra, namely a vertex Lie algebra. On the other hand, it is known that associated to every conformal Lie algebra C one has an authentic Lie algebra CLie, and furthermore has a canonical vertex algebra VC with U(CLie) as its underlying space. We show that VC is a connected cocommutative vertex bialgebra. On the other hand, for a general vertex bialgebra V , we show that the set P(V ) of primitive elements is a vertex Lie algebra. Furthermore, we show that if V is a connected cocommutative vertex bialgebra, then V is isomorphic to the vertex bialgebra VP(V ) associated to the vertex Lie algebra P(V ). In particular, this shows that every cocommutative connected vertex bialgebra V is isomorphic to VP(V ) and hence establishes the equivalence between the category of cocommutative connected vertex bialgebras and the category of vertex Lie algebras. This talk is partially based on a joint work with Jianzhi Han and Yukun Xiao.