In the determination of the bioavailability of drugs administered orally, the drugs’ solubility and permeability play a crucial role. For absorption of drug molecules and production of a pharmacological response, solubility is an important parameter that defines the concentration of the drug in systemic circulation. It is a challenging task to improve the oral bioavailability of drugs that have poor water solubility. Most drug molecules are either poorly soluble or insoluble in aqueous environments. Polymer nanocomposites are combinations of two or more different materials that possess unique characteristics and are fused together with sufficient energy in such a manner that the resultant material will have the best properties of both materials. These polymeric materials (biodegradable and other naturally bioactive polymers) are comprised of nanosized particles in a composition of other materials. A systematic search was carried out on Web of Science and SCOPUS using different keywords, and 485 records were found. After the screening and eligibility process, 88 journal articles were found to be eligible, and hence selected to be reviewed and analyzed. Biocompatible and biodegradable materials have emerged in the manufacture of therapeutic and pharmacologic devices, such as impermanent implantation and 3D scaffolds for tissue regeneration and biomedical applications. Substantial effort has been made in the usage of bio-based polymers for potential pharmacologic and biomedical purposes, including targeted deliveries and drug carriers for regulated drug release. These implementations necessitate unique physicochemical and pharmacokinetic, microbiological, metabolic, and degradation characteristics of the materials in order to provide prolific therapeutic treatments. As a result, a broadly diverse spectrum of natural or artificially synthesized polymers capable of enzymatic hydrolysis, hydrolyzing, or enzyme decomposition are being explored for biomedical purposes. This summary examines the contemporary status of biodegradable naturally and synthetically derived polymers for biomedical fields, such as tissue engineering, regenerative medicine, bioengineering, targeted drug discovery and delivery, implantation, and wound repair and healing. This review presents an insight into a number of the commonly used tissue engineering applications, including drug delivery carrier systems, demonstrated in the recent findings. Due to the inherent remarkable properties of biodegradable and bioactive polymers, such as their antimicrobial, antitumor, anti-inflammatory, and anticancer activities, certain materials have gained significant interest in recent years. These systems are also actively being researched to improve therapeutic activity and mitigate adverse consequences. In this article, we also present the main drug delivery systems reported in the literature and the main methods available to impregnate the polymeric scaffolds with drugs, their properties, and their respective benefits for tissue engineering.