When randomized control trials are infeasible, we rely on observational studies to determine causal effects of treatments. The control and treated groups in observational studies are not immediately comparable due to selection bias. This thesis presents new stratification and matching methods that can be used to select subsets or weighted versions of the groups that are comparable based on measured covariates. The first method, natural stratification, is presented in Chapters 2 and 3. This method stratifies based on several important covariates and then discards some control units such that there is the same ratio of treated to control units within each stratum and the remaining units are comparable in terms of all measured covariates. Chapter 2 compares the risk of stillbirth in older mothers to that in younger mothers. Chapter 3 compares the risk of central or peripheral nervous system side effects when using fluoroquinolones to treat sinus infections as opposed to azithromycin or amoxicillin. The second method, optimal refinement of strata, is presented in Chapter 4. This method looks at existing stratifications, such as those based on the propensity score, and splits each stratum into two refined strata with improved covariate balance, addressing any residual differences between the control and treated groups in the original stratification without discarding any units. This is illustrated with a reanalysis of whether right heart catheterization, an invasive technique used to guide treatment for critically ill patients, is harmful. The final method, triples matching, is presented in Chapter 5, which looks at whether playing high school football increases the risk of cognitive decline in old age. Either two controls and one treated unit or two treated units and one control are matched together based on covariate values. It is a simple design, with just two weights, that has desirable properties in terms of feasibility, ability to balance covariates, and insensitivity to unmeasured biases. A discussion of the benefits of each of the new methods is in Chapter 6.