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Anomaly matching gives a powerful tool to constrain the possible behaviors of strongly interacting quantum systems. It has been developed in the context of continuous chiral symmetry of gauge theories, and recent progress on symmetry-protected topological phases elucidates that its applicability is much broader. In the first part, I give a review on these developments of anomaly matching, and try to understand it using quantum mechanical example.

In the second part, we consider applications of anomaly matching to QCD and also to spin systems. For the application to QCD, I would like to talk not only about the conventional anomaly matching by 't Hooft but also about the matching of discrete anomaly, and discuss their consequences. For the spin system, I would like to explain the connection between the anomaly matching and Haldane conjecture, and I will also consider its generalization to SU(N) spin systems.