Surface plasmon resonance is an optical phenomenon first discovered in 1902. The phenomenon has since had many applications, particularly in biosensing. In this paper, we review surface plasmon resonance-based biosensing, look at recent progress made in integrating quantum resources to develop surface plasmon resonance-based biosensors into a class of surface plasmon resonance biosensors commonly referred to as quantum surface plasmon resonance biosensors, and examine the advantages which quantum biosensors bring. We will review recent experimental and theoretical work showing that making use of quantum states of light offers a great enhancement in the precision of our biosensor, as they can go below the shot-noise limit (standard quantum limit) of precision in intensity noise detection. An overview of the surface plasmon resonance mechanism, its applications, and some limitations, as well as a report on recent research to address certain limitations and quantum-based surface plasmon resonance sensing, are provided.