The intrinsic correlations of galaxy shapes are a recognised contaminant to weak gravitational lensing measurements. Luminous red galaxies have been observed to point radially towards each other, likely as a consequence of the action of the tidal field of the large-scale structure on their shapes and orientations; blue, disc-like galaxies are thought to be subject to no detectable alignment. If these intrinsic alignments are not accounted for, cosmological constraints from future surveys could be severely biased. I will describe our exploration of the intrinsic alignment signal in cosmological hydrodynamical simulations and to connect it to current observations. I will also discuss how intrinsic alignments might become themselves a cosmological probe in the future, helping constrain the physics of inflation.