This dissertation is structured as a collection of five papers formatted as chapters. The first four papers provide modeling and methodological contributions in scheduling or investment problems in power systems using the adaptive robust optimization framework and modifications to the column-and-constraint-generation algorithm (CCGA). The first paper addresses the security-constrained short-term scheduling problem where automatic primary response is considered. A two-stage robust model is adopted, resulting in complex mixed-integer linear instances featuring binary variables associated with first- and second-stage decisions. A new tailored CCGA which explores the structure of the problem is devised. The second paper uses deep neural networks for learning the mapping of nodal demands onto generators set point for the first paper s model. Robust-based modeling approaches and the CCGA are used to enforce feasibility for the solution. This method results in important computational gains as compared to results of the first paper. The third paper proposes an adaptive data-driven approach for a two-stage robust unit commitment model, where the polyhedral uncertainty set is characterized directly from data, through the convex hull of a set of previously observed non-dispatchable generation profiles. The resulting problem is suitable for the exact CCGA. The fourth paper proposes an adaptive two-stage distributionally robust transmission expansion model incorporating long- and short-term uncertainties. A novel extended CCGA is devised to tackle distributionally robust subproblems. Finally, under a different and higher-level perspective, the fifth paper investigates the adequacy of systematic inducement prizes for fostering innovations in theoretical and computational aspects for various modern power systems challenges.