This text proposes an epistemological modeling of the mathematical defining activity, and highlights the links between defining and proving. The goals are also didactical, in order to design, to analyze and to diffuse new situations involving defining activities. The focus of this research and the methodology are developed in the first part. The second part of this text presents a critical analysis of the international existing research regarding the study of the mathematical defining activity. It underscores several points of convergence, but also the tensions and lacks both from an epistemological and a didactical point of view in the modeling and the analysis of defining processes. The third part develops an epistemological reference modeling of the mathematical defining activity, taking into account the dialectic between defining and proving. This modeling with epistemological roots is also based upon several experiments (at secondary level and university level) and upon interviews with mathematicians. Theoretical frameworks from the didactic of mathematics (the model of conceptions among others) and mathematics (complexity theory) are called upon. The fourth and last part deals with new research perspectives from three points of view: epistemological, theoretical, and didactical/educational.
