JuMP is an open-source algebraic modeling language in the Julia language. In this work, we discuss a complete re-write of JuMP based on a novel abstract data structure, which we call \textit{MathOptInterface}, for representing instances of mathematical optimization problems. MathOptInterface is significantly more general than existing data structures in the literature, encompassing, for example, a spectrum of problems classes from integer programming with indicator constraints to bilinear semidefinite programming. We highlight the challenges that arise from this generality, and how we overcame them in the re-write of JuMP.