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In the course of classifying the homogeneous permutations, Cameron introducedthe viewpoint of permutations as structures in a language of two linear orders,and this structural viewpoint is taken up here. The majority of this thesis isconcerned with Cameron's problem of classifying the homogeneous structures in alanguage of finitely many linear orders, which we call finite-dimensionalpermutation structures. Towards this problem, we present a construction that weconjecture produces all such structures. Some evidence for this conjecture isgiven, including the classification of the homogeneous 3-dimensionalpermutation structures.  We next consider the topological dynamics, in the style of Kechris, Pestov,and Todor\v{c}evi\'c, of the automorphism groups of the homogeneousfinite-dimensional permutation structures we have constructed, which requiresproving a structural Ramsey theorem for all the associated amalgamationclasses. Because the $\emptyset$-definable equivalence relations in thesehomogeneous finite-dimensional permutation structures may form arbitrary finitedistributive lattices, the model-theoretic algebraic closure operation maybecome quite complex, and so we require the framework recently introduced byHubi\v{c}ka and Ne\v{s}etril.  Finally, we consider the decision problem for whether a finitely-constrainedpermutation avoidance class is atomic, or equivalently, has the joint embeddingproperty. As a first approximation to this problem, we prove the undecidabilityof the corresponding decision problem in the category of graphs. Modifying thisproof also gives the undecidability, in the category of graphs, of thecorresponding decision problem for the joint homomorphism property, which is ofinterest in infinite-domain constraint satisfaction problems.

Infinite Limits of Finite-Dimensional Permutation Structures, and their Automorphism Groups: Between Model Theory and Combinatorics