Many novel quantum effects can be observed in some special materials, which are the carriers of quantum effects and are called quantum materials. The research on quantum materials is leading a new technological revolution and is a hot frontier in many research fields. Because of the correlation between electrons and electrons in low-dimensional materials and the obvious limiting effect of electrons, there are rich and wonderful quantum effects or behaviors in low-dimensional materials. This paper mainly introduces several mechanisms affecting the basic properties of quantum materials from the quantum effects in materials, and focuses on the development history, development status and current challenges of several low-dimensional quantum materials (topological insulators, graphene, silene, germanene).