The three notions ``essential singularity of the i-th kind, \(i=1,2,3''\) of a meromorphic mapping are introduced: A meromorphic mapping \(f: X\to Y\), where \(X^ *\), Y are normal complex spaces and \(X\subset X^ *\) is an open subset, is said to have an essential singularity of the i-th kind in a point P lying on the border of X if f doesn't admit certain extension properties in a neighbourhood of P. Then, some existence theorems for meromorphic mappings with special sets of essential singularities are given: Among others it is shown that for any arbitrary closed subset \(X^ *\setminus X\) of \(X^ *\) there exists a meromorphic mapping \(f: X\to Y\) which in every point of the border of X has essential singularities of the first kind. This result is also interesting in connection with extention problems of meromorphic mappings into thin exceptional sets, which e.g. were examined by Karl Stein. At last, some theorems are given which can be helpful when trying to prove that a meromorphic mapping in a given point has no essential singularity.