Cold atom interferometers use laser-cooled atomic ensembles as test masses and convert gravity-induced phase shifts into absolute acceleration measurements. Their low drift, absence of mechanical wear, and capability for continuous observation make them suitable for precision gravity metrology, static gravity monitoring, and dynamic gravity measurement on moving platforms. This review introduces the basic principle and technical characteristics of cold atom interferometric absolute gravimetry. It summarizes recent progress in gravity metrology and metrological traceability, ground static measurement, transportable and field measurement, shipborne measurement, and airborne measurement. It also discusses prospective underwater and spaceborne applications. The review compares the system configurations, error correction methods, vibration compensation strategies, data processing procedures, and performance evaluation methods used in different application scenarios. It further summarizes engineering progress from the AVIC Changcheng Institute of Metrology & Measurement in quantum absolute gravity traceability, validation of static quantum gravimeters, shipborne dynamic measurement, and airborne dynamic measurement. Current developments show that cold atom absolute gravimeters are moving from high-precision laboratory instruments toward metrological, product-oriented, and multi-platform engineering applications. However, further improvements are still needed in domestic core components, long-term reliable operation, environmental adaptability, dynamic compensation accuracy, data fusion, and evaluation standards. Future work should coordinate the development of core devices, complete instruments, platform adaptation, data processing, and metrological evaluation for stationary stations, field surveys, vehicle-borne measurements, shipborne measurements, airborne measurements, underwater platforms, and spaceborne missions. These developments will support gravity reference maintenance, geophysical exploration, marine gravity surveys, and airborne gravity measurement.