Perhaps the best known use of modern techniques for optimization in observational studies is within matching algorithms, wherein treated units are placed into matched sets with similar control units to adjust for overt biases. While the intuitive appeal of matching has been long understood, its ascent in popularity can be attributed in large part to computational advances in network flow optimization. This dissertation explores how modern optimization can be leveraged to address other problems in observational studies. First, we demonstrate how, in the absence of covariate overlap, the maximal box problem can be used to define an interpretable study population wherein inference can be conducted without extrapolating on important variables. Next, we discuss how integer programming can be used to perform inference, construct confidence intervals, and provide sensitivity analyses for meaningful causal estimands in matched observational studies when the outcomes of interest are binary. Third, we present a method utilizing convex optimization for conducting a sensitivity analysis when there are multiple outcome variables of interest which, we show, can help attenuate the loss in power from accounting for multiple comparisons when assessing the robustness of a study's findings to unmeasured confounding. Finally, we present methods for conducting a sensitivity analysis for the average treatment effect with continuous outcome variables with and without assuming a known direction of effect.