Conway's groupoid and its relatives

In 1987, John Horton Conway constructed a subset M-13 of permutations on a set of size 13 for which the subset fixing any given point is isomorphic to the Mathieu group M-12. The construction has fascinated mathematicians for the past thirty years, and remains remarkable in its mathematical isolation. It is based on a "moving-counter puzzle" on the projective plane PG(2, 3). This survey, a homage to John Conway and his mathematics, discusses consequences and generalisations of Conway's construction. In particular it explores how various designs and hypergraphs can be used instead of PG(2, 3) to obtain interesting analogues of M-13. In honour of John Conway, we refer to these analogues as Conway groupoids. A number of open questions are presented.