  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.
