In our previous works, we introduced many different derivation Lie algebras to isolated singularities. We investigated their relations in the case of weighted homogeneous singularities. Moreover, explicit formulas of the dimensions of these of Lie algebras were given for fewnomial singularities. Several conjectures were proposed and verified partially. In this survey paper, we are going to summarize the definitions and results of the theory of derivation Lie algebras of isolated singularities obtained in the past decade.