We present recent advances in using mixed-integer programming (MIP) to solve difficult nonconvex optimization problems arising in application areas such as operations, robotics, and power systems. The central methodological contributions of this work are two frameworks for constructing formulations that take advantage of problem structure: either geometric, or combinatorial. These techniques allow us to design novel MIP formulations that are small, strong, and have other desirable properties. We present computational results show ing that these new formulations outperform existing approaches, sometimes substantially (e.g. solving to optimality in orders of magnitude less time). Finally, we highlight how high-level, easy-to-use computational modeling tools developed in tandem, such as the JuMP modeling language and extensions to it, can help make these advanced formulations accessible to practitioners and researchers.