Droplet impact on a flexible substrate is a prevalent phenomenon in nature and various advanced technologies such as soft bio-printing, tissue engineering, smart biomaterials and flexible electronics. Recent rapid advancement in new functional surfaces, ultra-high-speed imaging, nanotechnology, deep learning, advanced computational strength and the relation between fluid dynamics and interfacial science have intensified the physical understanding of droplet impact on soft materials. Once a droplets impacts on a solid surface, it deposits, spreads, rebounds or splashes. Given the importance of the droplet impact onto soft substrates in biotechnology, medicine and advanced flexible electronics, a deep physical understanding of such complex phenomenon is vital. This review initially presents the liquid-solid interaction physics and relevant interfacial science. Next, this review discusses the physics of droplet impact on soft materials with different physical and interfacial characteristics. Moreover, this review presents advancements in droplet impact on elastic materials relevant to new technologies such as soft electronics, elastic smart biomaterials, tissue engineering and the fight against COVID-19 pandemic. Finally, this review lays out future research directions related to current problems in such complex physical phenomenon.