  We give several unequivalent notions of convergency of meromorphic functions
and more generally meromorphic mappings (strong, weak, $\Gamma $-convergency
and some others). Relations between them are investigated. A version of Rouche
theorem for the strongly converging sequences of meromorphic mappings is given.
  Further we investigate the sets of normality of families of meromorphic
mappings, connections with extension properties of those mappings, some apriori
estimates for the volume. Finally we apply the technique developped to the
study of Fatou sets of meromorphic endomorphisms of K\"ahler surfaces.
