An antichain in a (partially) ordered set X is a subset with no comparable pairs, and a fibre in X is a subset that intersects every maximal antichain. There are well-known combinatorial problems that involve enumeration of antichains, such as Dedekind’s problem. Somewhat more recent questions concern enumeration of maximal antichains and estimates of minimum fibre sizes in ordered sets overall, and for special classes such as subset lattices. In this talk, we will present results and problems of these types, along with some quite natural graph and hypergraph analogues.