This paper addresses the issue of limited overlap in covariate distributions when estimating average treatment effects (ATE) under unconfoundedness or exogenous treatment assignment. Limited overlap can lead to imprecise estimates and make commonly used estimators sensitive to specification choices. The authors propose a systematic approach to dealing with this issue by characterizing optimal subsamples for which the ATE can be estimated most precisely, as well as optimally weighted average treatment effects. Under certain conditions, the optimal selection rules depend solely on the propensity score. For a wide range of distributions, a good approximation to the optimal rule is provided by dropping all units with estimated propensity scores outside the range [0.1, 0.9]. The paper also discusses the properties of estimators for these optimal estimands and provides numerical calculations based on the Beta distribution. The authors illustrate their methods using data from a non-experimental study on labor market programs, demonstrating that their approach can significantly reduce the variance of estimates.This paper addresses the issue of limited overlap in covariate distributions when estimating average treatment effects (ATE) under unconfoundedness or exogenous treatment assignment. Limited overlap can lead to imprecise estimates and make commonly used estimators sensitive to specification choices. The authors propose a systematic approach to dealing with this issue by characterizing optimal subsamples for which the ATE can be estimated most precisely, as well as optimally weighted average treatment effects. Under certain conditions, the optimal selection rules depend solely on the propensity score. For a wide range of distributions, a good approximation to the optimal rule is provided by dropping all units with estimated propensity scores outside the range [0.1, 0.9]. The paper also discusses the properties of estimators for these optimal estimands and provides numerical calculations based on the Beta distribution. The authors illustrate their methods using data from a non-experimental study on labor market programs, demonstrating that their approach can significantly reduce the variance of estimates.