The moduli space of marked singularities parameterizes mu-homotopic isolated hypersurface singularities equipped with certain markings. This moduli space can be understood either as a global mu-constant stratum or as a Teichmüller space of singularities. The additional marking allows one to formulate the conjecture on the analytic behavior of singularities within a distinguished mu-homotopy class in terms of a Torelli type problem in an efficient way. In my talk I will discuss the history of this problem and introduce carefully the notion of a marked singularity.