Archaeological prospecting is pivotal to understanding human civilization's development and past human-environment interactions (PHEIs). Synthetic aperture radar (SAR), a unique active remote sensing (RS) technology capable of penetrating vegetation and dry soil, has become an indispensable tool for large-scale archaeological surveys. This review, encompassing 45 years of progress (1980-2024), systematically chronicles SAR's evolution-a field we term SARchaeology-from pioneering L-band discoveries to sophisticated modern techniques like high-resolution SAR (HiSAR), interferometric SAR (InSAR), polarimetric SAR (PolSAR), and multitemporal SAR analysis (MtSAR). These methods have enabled unprecedented discoveries of hidden landscapes, particularly in areas obscured by dense cover or aridity. However, SARchaeology grapples with inherent limitations we define as the "SAR gap," driven by technical complexity and, critically, the difficulty in achieving adequate signal-to-noise ratios (SNR) for small low-backscatter archaeological targets. Traditional challenges like speckle noise, cost, and interpretation ambiguity are compounded by this fundamental issue. To bridge this gap, we propose an integrated framework of five strategic frontiers: 1) leveraging innovative long-wavelength SAR technologies (e.g., the P band) for deep penetration; 2) establishing collaborative space-air-ground networks; 3) integrating SAR with complementary RS techniques; 4) implementing artificial intelligence (AI) and machine learning (ML) for noise suppression and weak signal feature extraction; and 5) developing explainable SARchaeology (X-SARchaeology). This framework provides a definitive road map for the systematic exploration of buried civilizations in hyper-arid regions (e.g., the Silk Road and ancient Mesopotamia). By focusing on these solutions, SARchaeology is poised to transcend conventional limitations and illuminate much more of human history before its Golden Jubilee.