Since the discovery of the two-dimensional (2D) carbon material, graphene, just over a decade ago, the development of graphene-based field effect transistors (G-FETs) has become a widely researched area, particularly for use in point-of-care biomedical applications. G-FETs are particularly attractive as next generation bioelectronics due to their mass-scalability and low cost of the technology’s manufacture. Furthermore, G-FETs offer the potential to complete label-free, rapid, and highly sensitive analysis coupled with a high sample throughput. These properties, coupled with the potential for integration into portable instrumentation, contribute to G-FETs’ suitability for point-of-care diagnostics. This review focuses on elucidating the recent developments in the field of G-FET sensors that act on a bioaffinity basis, whereby a binding event between a bioreceptor and the target analyte is transduced into an electrical signal at the G-FET surface. Recognizing and quantifying these target analytes accurately and reliably is essential in diagnosing many diseases, therefore it is vital to design the G-FET with care. Taking into account some limitations of the sensor platform, such as Debye–Hükel screening and device surface area, is fundamental in developing improved bioelectronics for applications in the clinical setting. This review highlights some efforts undertaken in facing these limitations in order to bring G-FET development for biomedical applications forward.