In the Lambda-CDM cosmological paradigm, structure grows hierarchically, making mergers an essential ingredient of any galaxy formation model. In particular, a precise determination of the galaxy merger rate is crucial in order to bring galaxy formation models into agreement with the observed distribution of galaxy morphologies, as well as to explain the frequency of starburst galaxies and active galactic nuclei. Although the merger rate of dark matter halos is relatively well understood, with various theoretical estimates differing by less than 50%, similar agreement has yet to be achieved for the galaxy merger rate, for which different theoretical predictions show a scatter of up to an order of magnitude. We use simulations from the Illustris project, a suite of large-scale hydrodynamical simulations carried out with the novel moving-mesh code AREPO, to construct merger trees and calculate the *galaxy* merger rate as a function of descendant galaxy mass, progenitor mass ratio, and redshift. We find that the galaxy merger rate has a simple dependence on these three parameters. Our results are in stark contrast with previous studies based on semi-analytic models and hydrodynamical simulations of galaxy formation, but are in relatively good agreement with predictions from semi-empirical models, which estimate the galaxy merger rate through a combination of N-body (dark matter-only) simulations and observations, but without attempting to model galaxy formation from first principles.