Generalized models provide a framework for the study of evolution equations without specifying all functional forms in the defining vector field. The generalized formulation of problems has been shown to facilitate the analytical investigation of local dynamics and has been used successfully to answer applied questions. Yet their potential to facilitate analytical computations has not been realized in the mathematical literature. In the present paper, we introduce the method of generalized modelling in mathematical terms, supporting the key steps of the procedure by rigorous proofs. Further, we point out open questions that are in the scope of present mathematical research and if answered could greatly increase the predictive power of generalized models.