: One of the important subjects that has been recently considered in operator theory is the characterization of linear operators that preserve certain relation or properties between operators. In this thesis we explain what we mean by the word " preserving" and we give the characterization of the linear operators that preserve nilpotency , unitary similarity , asympotic similarity and similarity. The investigation of the conditions under which preserving asympotic similarity and up the preserving similarity is another interesting subject that will also be considered in this thesi