To develop further theory of limit theorems which based on the concepts of convergence in distribution or in probability of r.v’s

some authors intruduced anew type of convergence of sequences of r.v’s--stable convergence. We discuss this convergence more universally with sequences of random elements on a general complete separable metric space. and get some new results. Moreover. we make clear the relations between stable convergence and convergence in distribution or in probability. and give some interesting examples.