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We consider causal inference in two-sided matching markets, particularly in a school choice context, where the researcher is interested in understanding the treatment effect of schools on students. We characterize two classes of mechanisms that can be considered natural experiments, simulable mechanisms and cutoff mechanisms, which are mathematically general and encompass a large set of allocation mechanisms used in practice. We propose estimation and inference procedures for causal effects given each of these mechanisms, and characterize the statistical properties of the resulting causal estimators. Our approach allows us to relax the simplifying large-market assumption made in earlier work (Abdulkadiroğlu et al., 2017a, 2019), and we show that classical regression discontinuity procedures extend to settings where the discontinuity cutoff is endogenously chosen. Our results provide a rigorous statistical basis for causal inference and program evaluation in a number of settings where treatment assignment is complex.