© 2016 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society.We investigate correlations between different physical properties of star-forming galaxies in the 'Evolution and Assembly of GaLaxies and their Environments' (EAGLE) cosmological hydrodynamical simulation suite over the redshift range 0 = z = 4.5. A principal component analysis reveals that neutral gas fraction (fgas,neutral), stellar mass (Mstellar) and star formation rate (SFR) account for most of the variance seen in the population, with galaxies tracing a two-dimensional, nearly flat, surface in the three-dimensional space of fgas, neutral-Mstellar-SFR with little scatter. The location of this plane varies little with redshift, whereas galaxies themselves move along the plane as their fgas, neutral and SFR drop with redshift. The positions of galaxies along the plane are highly correlated with gas metallicity. The metallicity can therefore be robustly predicted from fgas, neutral, or from the Mstellar and SFR. We argue that the appearance of this 'Fundamental Plane of star formation' is a consequence of self-regulation, with the plane's curvature set by the dependence of the SFR on gas density and metallicity. We analyse a large compilation of observations spanning the redshift range 0 ? z ? 3, and find that such a plane is also present in the data. The properties of the observed Fundamental Plane of star formation are in good agreement with EAGLE's predictions.