The majority of astrophysics involves the study of spiral galaxies, and stars and planets within them, but how spiral arms in galaxies form and evolve is still a fundamental problem. Major progress in this field was made primarily in the 1960s, and early 1970s, but since then there has been no comprehensive update on the state of the field. In this review, we discuss the progress in theory, and in particular numerical calculations, which unlike in the 1960s and 1970s, are now commonplace, as well as recent observational developments. We set out the current status for different scenarios for spiral arm formation, the nature of the spiral arms they induce, and the consequences for gas dynamics and star formation in different types of spiral galaxies. We argue that, with the possible exception of barred galaxies, spiral arms are transient, recurrent and initiated by swing amplified instabilities in the disc. We suppose that unbarred m = 2 spiral patterns are induced by tidal interactions, and slowly wind up over time. However the mechanism for generating spiral structure does not appear to have significant consequences for star formation in galaxies.