This thesis explores mathematical optimization techniques to address the computeraided molecular and mixture design problems (CAMD/CAMxD). In particular, we leverage the power of mixed-integer linear programs (MILPs) to quickly and efficiently design over the massive chemical search space. These MILPs, when coupled with state-ofthe- art derivative-free optimization (DFO) methods, make for an efficient optimization strategy when designing mixtures of molecules or when considering a single molecule design problem that involves difficult thermodynamics or process models. In the first chapter, we provide a very general overview of the field of CAMD as addressed from the perspective of mathematical optimization. We discuss many relevant quantitative structure-property relationships (QSPRs) and provide constraints typically used in CAMD/CAMxD optimization problems. The second chapter introduces our DFO-based molecular/mixture design algorithm and describes how this approach enables a much greater molecular diversity to be considered in the search space as compared to traditional methods. Additionally, this chapter looks at a few case studies relevant to crystallization solvents and provides a detailed comparison of 27 different DFO algorithms for solving these problems. The third chapter introduces COSMO-RS/-SAC as alternatives to UNIFAC as the method used to capture mixture thermodynamics for a variety of CAMD/CAMxD problems. To fully incorporate COSMO-RS/-SAC into CAMD, we introduce group contribution (GC) methods for estimating a few necessary parameters for COSMO-based methods. We demonstrate the utility of COSMO-RS/-SAC in a few case studies for which UNIFAC-like methods are insufficient. In the fourth chapter, we investigate reaction solvent design using COSMO-based methods. COSMO-RS is particularly suitable for these problems as they allow for modeling of many relevant species in chemical reactions (transition states, charges, etc.) directly at the quantum level. This information can be immediately passed to the CAMD problem. We investigate a number of solvent design problems for a few difficult reactions. We summarize the work and provide a few future directions in the final chapter. Overall, this thesis serves to push the field of CAMD forward by introducing new methods to more efficiently explore the massive chemical search space and to enable a few new classes of problems which were previously untenable.