Propensity score (PS) matching is widely used for studying treatment effects in observational studies. This article proposes the method of matching weights (MWs) as an analog to one-to-one pair matching without replacement on the PS with a caliper. Compared with pair matching, the proposed method offers more efficient estimation, more accurate variance calculation, better balance, and simpler asymptotic analysis. A statistical test for the misspecification of the PS model is proposed for balance checking purposes. An augmented version of the MW estimator is developed that has the double robust property, that is, the estimator is consistent, if either the outcome model or the PS model is correct. The proposed method is studied in simulations and illustrated through a real data example.