Low dimensional systems have fascinated physicists for a long time due to their unusual properties such as charge fractionalization, semionic statistics, and Luttinger liquid behavior among others. In nature, however, low dimensional systems often suffer from thermal fluctuations that can make these systems structurally unstable. Human beings, however, can trick nature by producing artificial structures which are not naturally produced. This Colloquium reviews the problem of self-assembled atomic wires on solid surfaces from an experimental and theoretical point of view. These materials represent a class of one-dimensional systems with very unusual properties that can open doors to the study of exotic physics that cannot be studied otherwise.