Laurenz Hudetz (University of Salzburg) gives a talk at the Workshop on Five Years MCMP: Quo Vadis, Mathematical Philosophy? (2-4 June, 2016) titled "Towards an Adequate Criterion of Structural Equivalence of Theories". Abstract: My aim in this talk is to provide a general and adequate explication of structural equivalence of scientific theories. I will first give a brief overview of the recent debate about criteria for structural equivalence and highlight the main problems of the criteria proposed so far. I argue that an adequate criterion of equivalence should explicitly take into account morphisms between the models of theories. The criterion of categorical equivalence does this and has been frequently considered recently (cf. Weatherall, 2015; Barrett, Rosenstock and Weatherall, 2015; Hudetz, 2015; Halvorson, 2016; Barrett and Halvorson, 2016; Weatherall, 2016; Halvorson and Tsementzis, 2016). Yet, it is not free of problems. I show that categorial equivalence is much too wide as a criterion of structural equivalence of theories. Then I will propose a solution to this problem by specifying a strengthening of categorical equivalence, which I call 'definable categorical equivalence'. This strengthened criterion employs the model-theoretic notion of definability. I argue that definable categorical equivalence is neither too wide nor too narrow.