This paper studies the problem of uniqueness of meromorphic functions and shows that there exist two finite setsSj (j=1, 2) such that any two nonconstant meromorphic functions f and g satisfying Ef (Sj)=Eg (Sj) for j=1, 2 must be identical. The results in this paper improve some theorems given by Nevanlinna7, Brosch8, Yi6 and other authors. As a special case, these results answer an open question posed by Gross5.