The quantum anomalous Hall effect (QAHE) is a quantized Hall effect without external magnetic field

which has chiral edge states and can be used for developing low consumption electronic devices and constructing other novel quantum states. At the end of 2012

the QAHE was first observed by Chinese scientists in Cr-doped (Bi

Sb)2Te3 topological insulator thin films. In the past seven years

the observation temperature has been raised from 30 mK to about 2 K. Further raising observation temperature is essential for the applications of many quantum effects

which is one of the major research directions in the field of topological quantum physics and materials. In this review

we summarize the experimental progresses in the study of QAHE in magnetic topological insulators

especially in the aspect of QAHE observation temperature. The article consists of four parts: in the first two parts

the QAHE in doping and proximity-effect induced magnetic topological insulators is introduced. In the third part

the newly discovered intrinsic magnetic topological insulator system is presented. The last part gives some perspectives about possible principle and roadmap of how to design and build QAH systems at high temperatures.