A new method is presented for estimating crustal thickness from gravity and topography data on the Moon. By calculating analytically the exterior gravitational field for a set of arbitrarily shaped polyhedra, relief along the crust–mantle interface can be inverted for that satisfies the observational constraints. As this method does not rely upon filtering the Bouguer anomaly, which was required with previous inversions performed in the spherical-harmonic domain, and as the dramatic variations in spatial quality of the lunar gravity field are taken into account, our crustal thickness model more faithfully represents the available data. Using our model results, we investigate various aspects of the prominent nearside impact basins. The crustal thickness in the central portion of the Orientale and Crisium basins is found to be close to zero, suggesting that these basins could have conceivably excavated into the lunar mantle. Furthermore, given our uncertain knowledge of the density of the crust and mantle, it is possible that the Humorum, Humboldtianum, Nectaris, and Smythii basins could have excavated all the way through the crust as well. The crustal structure for most of the young impact basins implies a depth/diameter ratio of about 0.08 for their excavation cavities. As noted in previous studies, however, the crustal structure of Imbrium and Serenitatis is anomalous, which is conceivably a result of enhanced rates of post-impact viscous relaxation caused by the proximity of these basins to the Procellarum KREEP Terrane. Impact basins older than Smythii show little or no evidence for crustal thinning, suggesting that these ancient basins were also affected by high rates of viscous relaxation resulting from higher crustal temperatures early in the Moon's evolution. The lithosphere beneath many young basins is found to be supporting a downward directed force, even after the load associated with the mare basalts is removed, and this is plausibly attributed to superisostatic uplift of the crust–mantle interface. Those basins that are close to achieving a pre-mare isostatic state are generally found to reside within, or close to, the Procellarum KREEP Terrane.