The discovery of the quantum Hall effect (QHE) in two-dimensional electron systems under a magnetic field ushered in the paradigm of topological matter and electron fractionalization over forty years ago. Recent advances in topological and moire materials have brought into focus the search for QHE at zero magnetic field, also known as the quantum anomalous Hall effect (QAHE). The QAHE may occur in topological bands and requires magnetic ordering that spontaneously break time reversal symmetry.

In this talk, I will first review earlier works on the integer QAHE in magnetically doped topological insulator thin films. I will then describe the theoretical search and experimental discovery of both integer and fractional QAHE in twisted transition metal dichalcogenide bilayers. The phase diagram as a function of moire band filling is discussed, where a composite Fermi liquid state at half filling plays a central role.