During the past two decades, tremendous progress has been made in the development of biodegradable polymeric materials for various industrial applications, including human and veterinary medicine. They are promising alternatives to commonly used non-degradable polymers to combat the global plastic waste crisis. Among biodegradable polymers used, or potentially applicable to, veterinary medicine are natural polysaccharides, such as chitin, chitosan, and cellulose as well as various polyesters, including poly(ε-caprolactone), polylactic acid, poly(lactic-co-glycolic acid), and polyhydroxyalkanoates produced by bacteria. They can be used as implants, drug carriers, or biomaterials in tissue engineering and wound management. Their use in veterinary practice depends on their biocompatibility, inertness to living tissue, mechanical resistance, and sorption characteristics. They must be designed specifically to fit their purpose, whether it be: (1) facilitating new tissue growth and allowing for controlled interactions with living cells or cell-growth factors, (2) having mechanical properties that address functionality when applied as implants, or (3) having controlled degradability to deliver drugs to their targeted location when applied as drug-delivery vehicles. This paper aims to present recent developments in the research on biodegradable polymers in veterinary medicine and highlight the challenges and future perspectives in this area.