The latest trends in the field of the on-site detection of chemical warfare agents (CWAs) involve increasing the availability of point detectors to enhance the operational awareness of commanders and soldiers. Among the intensively developed concepts aimed at meeting these requirements, wearable detectors, gas analyzers as equipment for micro- and mini-class unmanned aerial vehicles (UAVs), and distributed sensor networks can be mentioned. One of the analytical techniques well suited for use in this field is surface acoustic wave sensors, which can be utilized to construct lightweight, inexpensive, and undemanding gas analyzers for detecting CWAs. This review focuses on the intensively researched and developed variant of this technique, utilizing absorptive sensor layers dedicated for nerve CWAs’ detection. The paper describes the mechanism of the specific interaction occurring between the target analyte and the sensing layer, which serves as the foundation for their selective detection. The main section of this paper includes a chronological review of individual achievements in the field, largely based on the peer-reviewed scientific literature dating back to the mid-1980s to the present day. The final section presents conclusions regarding the prospects for the development of this analytical technique in the targeted application.