Stratification and rerandomization are two well-known methods used in randomized experiments for balancing the baseline covariates. Renowned scholars in experimental design have recommended combining these two methods; however, limited studies have addressed the statistical properties of this combination. This paper proposes two rerandomization methods to be used in stratified randomized experiments, based on the overall and stratum-specific Mahalanobis distances. The first method is applicable for nearly arbitrary numbers of strata, strata sizes, and stratum-specific proportions of the treated units. The second method, which is generally more efficient than the first method, is suitable for situations in which the number of strata is fixed with their sizes tending to infinity. Under the randomization inference framework, we obtain the asymptotic distributions of estimators used in these methods and the formulas of variance reduction when compared to stratified randomization. Our analysis does not require any modeling assumption regarding the potential outcomes. Moreover, we provide asymptotically conservative variance estimators and confidence intervals for the average treatment effect. The advantages of the proposed methods are exhibited through an extensive simulation study and a real-data example.