We present a new general theory of function-based hypergraph transformations on finite families of finite hypergraphs. A function-based hypergraph transformation formalises the action of structurally modifying hypergraphs from a family in a consistent manner. The mathematical form of the transformations facilitates their analysis and incorporation into larger mathematical structures, and concurs with the function-based nature of modelling in the physical world. Since quotients of hypergraphs afford their simplification and comparison, we also discuss the notion of a quotient hypergraph transformation induced by an equivalence relation on the vertex set of a hypergraph family. Finally, we demonstrate function-based hypergraph transformations with two fundamental classes of examples involving the addition or deletion of hyperedges or hypergraphs.