Book excerpt: [Formulae and special characters in this field can only be approximated. See PDF version for accurate reproduction] In this thesis we investigate the properties of k-closures of certain finite permutation groups. Given a permutation group G on a finite set Ω, for k ≥ 1, the k-closure G(k) of G is the largest subgroup of Sym(Ω) with the same orbits as G on the set Ωk of k-tuples from Ω. The first problem in this thesis is to study the 3-closures of affine permutation groups. In 1992, Praeger and Saxl showed if G is a finite primitive group and k ≥ 2 then either G(k) and G have the same socle or (G(k),G) is known. In the case where the socle of G is an elementary abelian group, so that G is a primitive group of affine transformations of a finite vector space, the fact that G(k) has the same socle as G gives little information about the relative sizes of the two groups G and G(k). In this thesis we use Aschbacher’s Theorem for subgroups of finite general linear groups to show that, if G ≤ AGL(d, p) is an affine permutation group which is not 3-transitive, then for any point α ∈ Ω, Gα and (G(3) ∩ AGL(d, p))α lie in the same Aschbacher class. Our results rely on a detailed analysis of the 2-closures of subgroups of general linear groups acting on non-zero vectors and are independent of the finite simple group classification. In addition, modifying the work of Praeger and Saxl in [47], we are able to give an explicit list of affine primitive permutation groups G for which G(3) is not affine. The second research problem is to give a partial positive answer to the so-called Polycirculant Conjecture, which states that every transitive 2-closed permutation group contains a semiregular element, that is, a permutation whose cycles all have the same length. This would imply that every vertex-transitive graph has a semiregular automorphism. In this thesis we make substantial progress on the Polycirculant Conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no semiregular elements. Publications arising from this thesis are [17, 54].