Fractional quantum Hall (FQH) states with fractionalzed quasiparticles are exotic topologically ordered quantum states driven by strong correlation between particles. Since the first discovery in 1982 in two-dimensional electron gases penetrated by strong magnetic fields, FQH physics has become an attractive frontier of condensed matter physics. From the last year, several research groups have reported observations of FQH transport at zero magnetic field in moiré superlattices based on transition metal dichalcogenides (TMD) and graphene. Moreover, evidence of fractional quantum spin Hall effect was also reported in TMD moiré superlattices. These results demonstrate that moiré superlattices are ideal platforms to control band structure and interactions to realize fractionalized topological states without external magnetic fields. In this paper, we will briefly review the recent progress. We will also summarize the remaining challenges and discuss the possible future development in this field.