We associate elliptic affine Lie algebras with what are called vertex
$\C((z))$-algebras and their modules in a certain category. In the course, we
construct two families of Lie algebras closely related to elliptic affine Lie
algebras.
We give a summary of the theory of (weak) quantum vertex $\C((t))$-algebras
and the association of quantum affine algebras with (weak) quantum vertex
$\C((t))$-algebras.