This article presents current possibilities of using polyester-based materials in hard and soft tissue engineering, wound dressings, surgical implants, vascular reconstructive surgery, ophthalmology, and other medical applications. The review summarizes the recent literature on the key features of processing methods and potential suitable combinations of polyester-based materials with improved physicochemical and biological properties that meet the specific requirements for selected medical fields. The polyester materials used in multiresistant infection prevention, including during the COVID-19 pandemic, as well as aspects covering environmental concerns, current risks and limitations, and potential future directions are also addressed. Depending on the different features of polyester types, as well as their specific medical applications, it can be generally estimated that 25–50% polyesters are used in the medical field, while an increase of at least 20% has been achieved since the COVID-19 pandemic started. The remaining percentage is provided by other types of natural or synthetic polymers; i.e., 25% polyolefins in personal protection equipment (PPE).