  The purpose of this article is threefold. The first is to construct a
Nevanlina theory for meromorphic mappings from a polydisc to a compact complex
manifold. In particular, we give a simple proof of Lemma on logarithmic
derivative for nonzero meromorphic functions on $\mathbb{C}^l.$ The second is
to improve the definition of the non-integrated defect relation of H. Fujimoto
\cite{F2} and to show two theorems on the new non-integrated defect relation of
meromorphic maps from a closed submanifold of $\mathbb{C}^l$ to a compact
complex manifold. The third is to give a unicity theorem for meromorphic
mappings from a Stein manifold to a compact complex manifold.
