Laver tables and combinatorics

The Laver tables are finite combinatorial objects with a simple elementary definition, which were introduced by R. Laver from considerations of logic and set theory. Although these objects exhibit some fascinating properties, they seem to have escaped notice from the combinatorics community. My aim is to give a short introduction to this topic, presenting the definition and main properties and stating a few open problems, which should arouse the interest of combinatorialists.