The phenomenon of drop impact on solid substrates is highly prevalent, widely observed in nature, daily life, and industrial production process. In the majority of practical liquid droplet impact scenarios, the substrate can be approximated as an ideal rigid substrate. However, in some certain scenarios, solid substrates with a smaller Young's modulus or geometric structures characterized by thin walls or elongated forms undergo significant deformation upon being impacted by liquid droplets. At this point, such an approximation is no longer applicable at all. The deformation of the solid substrate, in turn, affects the dynamics of the liquid droplet. Compared to the impact of liquid droplets on rigid substrates, the impact of liquid droplets on flexible substrates is a more complex fluid-structure coupling phenomenon, and this coupling interaction between liquid droplets and flexible substrates may give rise to new dynamic mechanisms, which may lead to different droplet dynamics. Therefore, the study of the impact dynamics of liquid droplets on flexible substrates is meaningful, and it has become one of the current research focuses of some scholars. Based on the geometric characteristics of flexible substrates, they can be categorized into three fundamental types: flexible bodies, flexible surfaces, and flexible rods. This article introduces relevant research progress on liquid droplet impact on flexible substrates according to this classification. The deformability of flexible substrates is the fundamental reason for the differences in the dynamics of liquid droplets on flexible substrates compared to the dynamics on rigid substrates. Existing researches show that substrate flexibility has an effect on various aspects of droplet dynamics, including contact time, maximum spreading factor, bubbles capture, drop rebound, and splashing. Some researchers have suggested that the impact of liquid droplets on appropriately flexible substrates can effectively reduce contact time and suppress splashing phenomena, which may offer new insights for manufacturing materials requiring waterproofing, anti-icing, and splash resistance.