According to general relativity, a clock experiencing a shift in the gravitational potential ΔU will measure a frequency change given by Δf/f ≈ ΔU/c2. The best clocks are optical clocks. After about 7 hr of integration they reach stabilities of Δf/f ∼ 10−18 and can be used to detect changes in the gravitational potential that correspond to vertical displacements of the centimetre level. At this level of performance, ground-based atomic clock networks emerge as a tool that is complementary to existing technology for monitoring a wide range of geophysical processes by directly measuring changes in the gravitational potential. Vertical changes of the clock's position due to magmatic, post-seismic or tidal deformations can result in measurable variations in the clock tick rate. We illustrate the geopotential change arising due to an inflating magma chamber using the Mogi model and apply it to the Etna volcano. Its effect on an observer on the Earth's surface can be divided into two different terms: one purely due to uplift (free-air gradient) and one due to the redistribution of matter. Thus, with the centimetre-level precision of current clocks it is already possible to monitor volcanoes. The matter redistribution term is estimated to be 3 orders of magnitude smaller than the uplift term. Additionally, clocks can be compared over distances of thousands of kilometres over short periods of time, which improves our ability to monitor periodic effects with long wavelength like the solid Earth tide.