Tissue engineering (TE) and regenerative medicine integrate information and technology from various fields to restore/replace tissues and damaged organs for medical treatments. To achieve this, scaffolds act as delivery vectors or as cellular systems for drugs and cells; thereby, cellular material is able to colonize host cells sufficiently to meet up the requirements of regeneration and repair. This process is multi-stage and requires the development of various components to create the desired neo-tissue or organ. In several current TE strategies, biomaterials are essential components. While several polymers are established for their use as biomaterials, careful consideration of the cellular environment and interactions needed is required in selecting a polymer for a given application. Depending on this, scaffold materials can be of natural or synthetic origin, degradable or nondegradable. In this review, an overview of various natural and synthetic polymers and their possible composite scaffolds with their physicochemical properties including biocompatibility, biodegradability, morphology, mechanical strength, pore size, and porosity are discussed. The scaffolds fabrication techniques and a few commercially available biopolymers are also tabulated.