The correlations of the morphological properties of galaxies with the surrounding large-scale structure are a unique probe of the influence of environment on galaxy formation and evolution processes. They can also generate limiting astrophysical systematics for cosmological measurements from galaxy surveys, in particular weak gravitational lensing. I will review the current state of the field regarding the understanding of alignment processes, their modelling, and observational results. I will also highlight the impact of intrinsic galaxy correlations on forthcoming surveys such as Euclid and outline measures that will be required to ensure accurate and precise constraints on cosmology.