Galaxy shapes are not randomly oriented, rather they are statistically
aligned in a way that can depend on formation environment, history and galaxy
type. Studying the alignment of galaxies can therefore deliver important
information about the physics of galaxy formation and evolution as well as the
growth of structure in the Universe. In this review paper we summarise key
measurements of galaxy alignments, divided by galaxy type, scale and
environment. We also cover the statistics and formalism necessary to understand
the observations in the literature. With the emergence of weak gravitational
lensing as a precision probe of cosmology, galaxy alignments have taken on an
added importance because they can mimic cosmic shear, the effect of
gravitational lensing by large-scale structure on observed galaxy shapes. This
makes galaxy alignments, commonly referred to as intrinsic alignments, an
important systematic effect in weak lensing studies. We quantify the impact of
intrinsic alignments on cosmic shear surveys and finish by reviewing practical
mitigation techniques which attempt to remove contamination by intrinsic
alignments.Comment: 52 pages excl. references, 16 figures; minor changes to match version
published in Space Science Reviews; part of a topical volume on galaxy
alignments, with companion papers arXiv:1504.05456 and arXiv:1504.0554