The outermost “crust” and an underlying, compositionally distinct, and denser layer, the “mantle,” constitute the silicate portion of a terrestrial planet. The “lithosphere” is the planet’s high-strength outer shell. The crust records the history of shallow magmatism, which along with temporal changes in lithospheric thickness, provides information on a planet’s thermal evolution. We focus on the basic structure and mechanics of Mercury’s crust and lithosphere as determined primarily from gravity and topography data acquired by the MESSENGER mission. We first describe these datasets: how they were acquired, how the data are represented on a sphere, and the limitations of the data imparted by MESSENGER’s highly eccentric orbit.  We review different crustal thickness models obtained by parsing the observed gravity signal into contributions from topography, relief on the crust–mantle boundary, and density anomalies that drive viscous flow in the mantle. Estimates of lithospheric thickness from gravity–topography analyses are at odds with predictions from thermal models, thus challenging our understanding of Mercury’s geodynamics. We show that, like those of the Moon, Mercury's ellipsoidal shape and geoid are far from hydrostatic equilibrium, possibly the result of Mercury's peculiar surface temperature distribution and associated buoyancy anomalies and thermoelastic stresses in the interior.