  We survey results regarding the definability and size of maximal discrete
sets in analytic hypergraphs. Our main examples include maximal almost disjoint
(or mad) families, $\mathcal I$-mad families, maximal eventually different
families, and maximal cofinitary groups. We discuss the non-increasing sequence
of cardinal characteristics $\mathfrak a_\xi$, for $\xi<\omega_1$ as well as
the notions of spectra of characteristics and optimal projective witnesses. We
give an account of Zhang's forcing to add generic cofinitary permutations, and
of a version of this forcing with built-in coding.
