  The search for topologically non-trivial states of matter has become an
important goal for condensed matter physics. Here, we give a theoretical
introduction to the quantum anomalous Hall (QAH) effect based on magnetic
topological insulators in two-dimension (2D) and three-dimension (3D). In 2D
topological insulators, magnetic order breaks the symmetry between the
counter-propagating helical edge states, and as a result, the quantum spin Hall
effect can evolve into the QAH effect. In 3D, magnetic order opens up a gap for
the topological surface states, and chiral edge state has been predicted to
exist on the magnetic domain walls. We present the phase diagram in thin films
of a magnetic topological insulator and review the basic mechanism of
ferromagnetic order in magnetically doped topological insulators. We also
review the recent experimental observation of the QAH effect. We discuss more
recent theoretical work on the coexistence of the helical and chiral edge
states, multi-channel chiral edge states, the theory of the plateau transition,
and the thickness dependence in the QAH effect.
