In this thesis we study set-theoretic definability of maximal objects originatingfrom various branches of mathematics, encompassing set theory, combinatorics,group theory, measure theory and operator algebras.In Part II, which is based on joint work with Asger Törnquist, we study definability of maximal almost disjoint families. With a simple tree derivative process, wefirst give a new proof of the classical theorem, due to Mathias, stating that thereare no infinite analytic maximal almost disjoint families. With small adjustments,the process can be carried out and terminates in LωCK1 , which proves that for everyinfinite Σ11 almost disjoint family A there is a ∆11 infinite subset x of ω such thatx ∩ z is finite for every z ∈ A. Our argument can be adapted to prove that ifאL[a]1 < א1, then there are no infinite Σ12[a] maximal almost disjoint families. Asmall modification of the derivative process can also be used to prove that underMA(κ) there are no infinite κ-Suslin maximal almost disjoint families.Part III is a reproduction of a preprint on definability of maximal cofinitarygroups, authored jointly with David Schrittesser. We give a construction of a closed(even Π01) set which freely generates an Fσ (even Σ02) maximal cofinitary group.In this isomorphism class, this is the lowest possible complexity of a maximalcofinitary group. Additionally, we discuss obstructions to potential constructionsof Gδ maximal cofinitary groups and introduce (maximal) finitely periodic groups.In Part IV, which is also a reproduction of a preprint, we study maximal ortho-gonal families. We begin by giving a new, short and elementary proof of a theoremby Preiss and Rataj, stating that there are no analytic maximal orthogonal familiesof Borel probability measures on a Polish space. In case when the underlying spaceis compact and perfect, we establish that the set of witnesses to non-maximalityis comeagre. The idea of our argument is based on the original proof by Preissand Rataj, but with significant simplifications. Our proof generalises to show thatunder MA + ¬CH there are no Σ12 maximal orthogonal families, that under PDthere are no projective maximal orthogonal families and that under AD there areno maximal orthogonal families at all. Finally, we introduce a notion of strongorthogonality for states on separable C*-algebras and generalise a theorem due toKechris and Sofronidis, stating that for every analytic orthogonal family of Borelprobability measures there is a product measure orthogonal to all measures in thefamily, to states on a certain class of C*-algebras.