Due to the benefits of the high sensitivity, real-time response, no labeling requirement, and good selectivity, fiber optic sensors based on surface plasmon resonance (SPR) have gained popularity in biochemical sensing in recent years. The current research on such sensors is hot in enhancing sensitivity, improving detection accuracy, and achieving the detection of biochemical molecules. The goal of this work is to present a thorough overview of recent developments in the optical fiber SPR biosensor research. Firstly, it explores the basic principles and sensing structures of optical fiber SPR biosensors, focusing on four aspects. Subsequently, this paper introduces three fiber optic surface plasmon biosensors: SPR, localized surface plasmon resonance (LSPR), and long-range surface plasmon resonance (LRSPR). Each concept is explained from the perspective of the basic principles of fiber optic SPR biosensors. Furthermore, a classification of fiber optic SPR biosensors in health monitoring, food safety, environmental monitoring, marine detection, and other applications is introduced and analyzed. Eventually, this paper summarizes the current research directions of SPR biosensors. Meanwhile, it provides a prospective outlook on how fiber optic SPR sensors will develop in the future.