The growing demand for sustainable materials has led to increased interest in biodegradable polymer fibers and nonwoven mats due to their eco-friendly characteristics and potential to reduce plastic pollution. This review highlights how mechanical properties influence the performance and suitability of biodegradable polymer fibers across diverse applications. This covers synthetic polymers such as polylactic acid (PLA), polyhydroxyalkanoates (PHAs), polycaprolactone (PCL), polyglycolic acid (PGA), and polyvinyl alcohol (PVA), as well as natural polymers including chitosan, collagen, cellulose, alginate, silk fibroin, and starch-based polymers. A range of fiber production methods is discussed, including electrospinning, centrifugal spinning, spunbonding, melt blowing, melt spinning, and wet spinning, with attention to how each technique influences tensile strength, elongation, and modulus. The review also addresses advances in composite fibers, nanoparticle incorporation, crosslinking methods, and post-processing strategies that improve mechanical behavior. In addition, mechanical testing techniques such as tensile test machine, atomic force microscopy, and dynamic mechanical analysis are examined to show how fabrication parameters influence fiber performance. This review examines the mechanical performance of biodegradable polymer fibers and fibrous mats, emphasizing their potential as sustainable alternatives to conventional materials in applications such as tissue engineering, drug delivery, medical implants, wound dressings, packaging, and filtration.