  The main aim of this article is to give some sufficient conditions for a
family of meromorphic mappings on a domain D in C^n into P^N(C) to be
meromorphically normal if they satisfy only some very weak conditions with
respect to moving hypersurfaces in P^N(C), namely that their intersections with
these moving hypersurfaces, which may moreover depend on the meromorphic maps,
are in some sense uniform. Our results generalise and complete previous results
in this area, especially the works of Fujimoto, Tu, Tu-Li, Mai-Thai-Trang and
the recent work of Quang-Tan.
