A cosmologically significant population of very luminous high-redshift galaxies has recently been discovered at submm wavelengths. Advances in submm detector technologies have opened this new window on the distant Universe. Here we discuss the properties of the high-redshift submm galaxies, their significance for our understanding of the process of galaxy formation, and the selection effects that apply to deep submm surveys. The submm galaxies generate a significant fraction of the energy output of all galaxies in the early Universe. We emphasize the importance of studying a complete sample of submm galaxies, and stress that because they are typically very faint in other wavebands, these follow-up observations are very challenging. Finally, we discuss the surveys that will be made using the next generation of submm-wave instruments under development.

86 pages, 23 figures. In press at Physics Reports. Higher-quality version of Figure 1 can be found in astro-ph/9911069. Replaced to match proofs, correct minor typos and update references