Recently, the quantum anomalous Hall effect has been realized in magnetic topological insulators in several groups. Here in this talk, we give a theoretical introduction to the QAH effect based on magnetic topological insulators. We will discuss more recent theoretical work on the coexistence of the helical and chiral edge states, multi-channel chiral edge states, and the theory of the plateau transition in the QAH. Some of the theoretical predictions are already borne out in recent experiment. We also discuss realistic materials for QAH in other materials beyond magnetic topological insulators, such as ferromagnetic insulator heterostructures with higher Curie temperature and larger band gap.

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