Discovered (or invented?) by Richard Laver in the 1990s, the tables that are now known as Laver tables are finite structures obeying the self-distributivity law x(yz)=(xy)(xz). Although their construction is totally explicit, some of their combinatorial properties are (so far) established only using unprovable set theoretical axioms, a quite unusual and paradoxical situation. We shall explain the construction of Laver tables, their connection with set theory, and their potential applications in low-dimensional topology via the recent computation of some associated cocycles.