Electrospinning has emerged as a versatile and accessible technology for fabricating polymer fibers, particularly for biological applications. Natural polymers or biopolymers (including synthetically derivatized natural polymers) represent a promising alternative to synthetic polymers, as materials for electrospinning. Many biopolymers are obtained from abundant renewable sources, are biodegradable, and possess inherent biological functions. This review surveys recent literature reporting new fibers produced from emerging biopolymers, highlighting recent developments in the use of sulfated polymers (including carrageenans and glycosaminoglycans), tannin derivatives (condensed and hydrolyzed tannins, tannic acid), modified collagen, and extracellular matrix extracts. The proposed advantages of these biopolymer-based fibers, focusing on their biomedical applications, are also discussed to highlight the use of new and emerging biopolymers (or new modifications to well-established ones) to enhance or achieve new properties for electrospun fiber materials.