The oxides of copper have attracted the attention of scientists already for more than hundred years. This fascination is fueled by many outstanding properties of the material, for example, a semiconducting behavior that led to the first diode fabricated in electronics, a pronounced excitonic response that stimulated an intense search for Bose-Einstein condensation, and a pivotal role in unconventional superconductivity. Despite this central position in past and present research activities, many aspects of copper oxides are not sufficiently understood to date. This applies in particular to their surface characteristics, where even fundamental questions, such as the energetically favored termination of low-index Cu2O and CuO planes, are still subject of debates. This review aims at addressing these deficiencies by compiling state-of-the-art knowledge of the surface science of copper oxides, and especially of cuprous oxide.