Physical disfigurement due to congenital defects, trauma, or cancer causes considerable distress and physical impairment for millions of people worldwide; impacting their economic, psychological and social wellbeing. Since 3000 B.C., prosthetic devices have been used to address these issues by restoring both aesthetics and utility to those with disfigurement. Internationally, academic and industry researchers are constantly developing new materials and manufacturing techniques to provide higher quality and lower cost prostheses to those people who need them. New advanced technologies including 3D imaging, modeling, and printing are revolutionizing the way prostheses are now made. These new approaches are disrupting the traditional and manual art form of prosthetic production which are laborious and costly and are being replaced by more precise and quantitative processes which enable the rapid, low cost production of patient-specific prostheses. In this two part review, we provide a comprehensive report of past, present and emerging soft-tissue prosthetic materials and manufacturing techniques. In this review, part A, we examine, historically, the ideal properts of a polymeric material when applied in soft-tissue prosthetics. We also detail new research approaches to target specific tissues which commonly require aesthetic restoration (e.g. ear, nose and eyes) and discuss both traditional and advanced fabrication methods, from hand-crafted impression based approaches to advanced manufactured prosthetics. We discuss the chemistry and related details of most significant synthetic polymers used in soft-tissue prosthetics in Part B. As advanced manufacturing transitions from research into practice, the five millennia history of prosthetics enters a new age of economic, personalized, advanced soft tissue prosthetics and with this comes significantly improved quality of life for the people affected by tissue loss.