Elastin is one of the main components of the extracellular matrix; it provides resistance and elasticity to a variety of tissues and organs of the human body, besides participating in cellular signaling. On the other hand, elastin-derived peptides are synthetic biopolymers with a similar conformation and structure to elastin, but these possess the advantage of solubility in aqueous mediums. Due to their biological activities and physicochemical properties, elastin and related peptides may be applied as biomaterials to develop diverse biomedical devices, including scaffolds, hydrogels, and drug delivery systems for tissue engineering. Likewise, the combination of elastin with natural or synthetic polymers has demonstrated to improve the mechanical properties of biomedical products and drug delivery systems. Here we comprehensively describe the physicochemical properties and physiological functions of elastin. Moreover, we offer an overview of the use of elastin and its derivative polymers as biomaterials to develop scaffolds and hydrogels for tissue engineering. Finally, we discuss some perspectives on the employment of these biopolymers to fabricate new biomedical products.