The quantum anomalous Hall (QAH) effect is a quantum Hall effect induced by spontaneous magnetization instead of an external magnetic field. The effect occurs in two-dimensional (2D) insulators with topologically nontrivial electronic band structure characterized by a non-zero Chern number. The QAH insulator can be realized in a ferromagnetic topological insulator (TI) film as the result of magnetically induced gap-opening at the Dirac surface states. With molecular beam epitaxy techniques, we have prepared thin films of magnetically doped (Bi,Sb)₂Te₃ TI with well-controlled composition, thickness and chemical potential, and obtained ferromagnetic insulator phase in them. In such magnetic TI films, we have experimentally observed the quantization of the Hall resistance at h/e² at zero field, accompanied by a considerable reduction in the dissipation of electron transport, which unambiguously demonstrate the occurrence of the QAH effect. The temperature, thickness and magnetic-doping-level dependences of the QAH effect have been systematically studied, which clarifies the roles of the band structure, electron localization and magnetic order in the effect and provides clues for obtaining the effect at a higher temperature. The experimental progresses in the QAH effect pave the ways for applications of dissipationless quantum Hall edge states in low-energy-consuming devices and for realizations of other novel quantum phenomena such as chiral topological superconductivity and axion electrodynamics.