Fractional quantum Hall (FQH) states with fractionalized 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. Since last year, FQH transport at zero magnetic field has been observed in moiré superlattices based on transition metal dichalcogenides (TMDs) and graphene. Furthermore, the evidence of fractional quantum spin Hall effect has also been reported in TMD moiré superlattices. These results demonstrate that moiré superlattices are an ideal platform for controlling band structures and interactions to realize fractionalized topological states without the intervention of external magnetic fields. In this paper, we will briefly review the recent research progress on fractionalized topological states in moiré superlattices, summarize the existing challenges, and discuss possible future development of this field.