Macaulay posets are posets for which there is an analogue of the classical Kruskal-Katona theorem for finite sets. These posets are of great importance in many branches of combinatorics and have numerous applications. We survey mostly new and also some old results on Macaulay posets, where the intention is to present them as pieces of a general theory. In particular, the classical examples of Macaulay posets are included as well as new ones. Emphasis is also put on the construction of Macaulay posets, and their relations to other discrete optimization problems.