Extension Properties of Meromorphic Mappings with Values in Non-Kahler Manifolds

We prove an analogue of E. Levi's Continuity Principle for meromorphic mappings with values in arbitrary compact complex manifolds in place of the Riemann sphere $\cc\pp^1$. The result is achieved by introducing a new extension method for meromorphic mappings. One of the corollaries reads as follows: If a compact complex surface $X$ is not "among the known ones" then for every domain $\Omega$ in a Stein surface every meromorphic mapping $f:\Omega \to X$ is in fact holomorphic and extends as a holomorphic mapping $\hat f:\hat D\to X$ of the envelope of holomorphy $\hat D$ of $D$ into $X$. In this last version also two examples of compact complex maniflds are described with meromoprhic mappings into these manifolds having thin but non-analytic singularity sets.