Abstract: The multilayer network describes either strong or weak coupling or connections among complex systems. In order to provide a more systematic and comprehensive understanding of the study for the robustness of multilayer networks that based on percolation theory, this paper reviews the effects of interdependency characteristics of cross-layer nodes, connectivity characteristics of intra-layer nodes, interdependency characteristics of intra-layer nodes, attack strategies on the cascading dynamics, and robustness of multilayer networks. Different from single-layer network, the multilayer network will collapse suddenly as a first-order percolation transition when suffered attacks. At the same time, the multilayer networks with strong heterogeneity of degree distribution are more vulnerable and exhibit multiple phase transitions or hybrid phase transitions. In addition, the mechanism of weak interdependence enables multilayer network to describe more details of complex systems, such as the heterogeneity of interdependency strength, the asymmetry of interdependency strength, and the topology of interdependency relations. These results show that the cascading dynamics of multilayer networks are more complicated than that of single-layer networks. Ignoring the interdependencies among complex systems may lead to overestimation of the robustness of complex networks or even a completely wrong understanding.