Biosensors capable of onsite and continuous detection of environmental and food pollutants and biomarkers are highly desired, but only a few sensing platforms meet the “2-SAR” requirements (sensitivity, specificity, affordability, automation, rapidity, and reusability). A fiber optic evanescent wave (FOEW) sensor is an attractive type of portable device that has the advantages of high sensitivity, low cost, good reusability, and long-term stability. By utilizing functional nucleic acids (FNAs) such as aptamers, DNAzymes, and rational designed nucleic acid probes as specific recognition ligands, the FOEW sensor has been demonstrated to be a general sensing platform for the onsite and continuous detection of various targets ranging from small molecules and heavy metal ions to proteins, nucleic acids, and pathogens. In this review, we cover the progress of the fluorescent FNA-based FOEW biosensor since its first report in 1995. We focus on the chemical modification of the optical fiber and the sensing mechanisms for the five above-mentioned types of targets. The challenges and prospects on the isolation of high-quality aptamers, reagent-free detection, long-term stability under application conditions, and high throughput are also included in this review to highlight the future trends for the development of FOEW biosensors capable of onsite and continuous detection.