Surface plasmon polaritons (SPPs) can be generated in graphene at frequencies in the mid-infrared to terahertz range, which is not possible using conventional plasmonic materials such as noble metals. Moreover, the lifetime and confinement volume of such SPPs are much longer and smaller, respectively, than those in metals. For these reasons, graphene plasmonics has potential applications in novel plasmonic sensors and various concepts have been proposed. This review paper examines the potential of such graphene plasmonics with regard to the development of novel high-performance sensors. The theoretical background is summarized and the intrinsic nature of graphene plasmons, interactions between graphene and SPPs induced by metallic nanostructures and the electrical control of SPPs by adjusting the Fermi level of graphene are discussed. Subsequently, the development of optical sensors, biological sensors and important components such as absorbers/emitters and reconfigurable optical mirrors for use in new sensor systems are reviewed. Finally, future challenges related to the fabrication of graphene-based devices as well as various advanced optical devices incorporating other two-dimensional materials are examined. This review is intended to assist researchers in both industry and academia in the design and development of novel sensors based on graphene plasmonics.