  Frenkel, Lepowsky, and Meurman constructed a vertex operator algebra (VOA)
associated to any even, integral, Euclidean lattice. In the language of
physics, these are examples of chiral conformal field theories (CFT). In this
paper, we define non-chiral vertex operator algebra and some associated
notions. We then give a construction of a non-chiral VOA associated to an even,
integral, Lorentzian lattice and construct their irreducible modules. We obtain
the moduli space of such modular invariant non-chiral CFTs based on even,
self-dual Lorentzian lattices of signature $(m,n)$ assuming the validity of a
technical result about automorphisms of the lattice. We finally show that
Narain conformal field theories in physics are examples of non-chiral VOA. Our
formalism helps us to identify the chiral algebra of Narain CFTs in terms of a
particular sublattice and give us the decomposition of its partition function
into sum of characters.
