Polyesters based on 2,5-furandicarboxylic acid (FDCA) are a new class of biobased polymers with enormous interest, both from a scientific and industrial perspective. The commercialization of these polymers is imminent as the pressure for a sustainable economy grows, and extensive worldwide research currently takes place on developing cost-competitive, renewable plastics. The most prevalent method for imparting these polymers with new properties is copolymerization, as many studies have been published over the last few years. This present review aims to summarize the trends in the synthesis of FDCA-based copolymers and to investigate the effectiveness of this approach in transforming them to a more versatile class of materials that could potentially be appropriate for a number of high-end and conventional applications.