Achievement Scalarizing Functions (ASFs) are a class of scalarizing functions for multiobjective optimization problems that have been successfully implemented in many applications due to their mathematical elegance and decision making utility. However, no formal proofs of the fundamental properties of ASFs have been presented in the literature. Furthermore, developments of ASFs, including the construction of new ASFs, do not acknowledge or make use of these properties. We fill this gap by formalizing the theory of ASFs and provide the basis for a novel generalization of ASFs to Bivariate Achievement Scalarizing Functions (BASFs).