Gravitational waves excite quadrupolar vibrations of elastic bodies. Monitoring these vibrations was one of the first concepts proposed for the detection of gravitational waves by Joseph Weber. At laboratory scale, these experiments became known as resonant-bar detectors, which form an important part of the history of GW detection. Due to the dimensions of these bars, the targeted signal frequencies were in the kHz range. It was also Weber who suggested to monitor vibrations of the Moon to search for gravitational waves in the mHz band. His Lunar Surface Gravimeter was deployed on the Moon in 1972 by the Apollo 17 crew. A design error made it impossible to carry out the intended search for GWs, but the idea remains intriguing. We propose a Lunar Gravitational-Wave Antenna (LGWA) based on Weber’s idea. The key component is a next-generation, high-sensitivity seismometer to be deployed on the Moon. LGWA would have rich scientific synergies with other GW detectors like LISA and ET and also with EM observatories. At the same time, it would serve as a high-precision geophysical station shedding light on the interior structure of the Moon, the mechanisms of moonquakes, and the Moon's formation history. Furthermore, its technology is of high relevance for ET, and other future terrestrial GW detectors, as part of a new generation of seismic isolation systems.