  The alignments between galaxies, their underlying matter structures, and the
cosmic web constitute vital ingredients for a comprehensive understanding of
gravity, the nature of matter, and structure formation in the Universe. We
provide an overview on the state of the art in the study of these alignment
processes and their observational signatures, aimed at a non-specialist
audience. The development of the field over the past one hundred years is
briefly reviewed. We also discuss the impact of galaxy alignments on
measurements of weak gravitational lensing, and discuss avenues for making
theoretical and observational progress over the coming decade.
