  In the course of classifying the homogeneous permutations, Cameron introduced
the 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 is
concerned with Cameron's problem of classifying the homogeneous structures in a
language of finitely many linear orders, which we call finite-dimensional
permutation structures. Towards this problem, we present a construction that we
conjecture produces all such structures. Some evidence for this conjecture is
given, including the classification of the homogeneous 3-dimensional
permutation 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 homogeneous
finite-dimensional permutation structures we have constructed, which requires
proving a structural Ramsey theorem for all the associated amalgamation
classes. Because the $\emptyset$-definable equivalence relations in these
homogeneous finite-dimensional permutation structures may form arbitrary finite
distributive lattices, the model-theoretic algebraic closure operation may
become quite complex, and so we require the framework recently introduced by
Hubi\v{c}ka and Ne\v{s}etril.
  Finally, we consider the decision problem for whether a finitely-constrained
permutation avoidance class is atomic, or equivalently, has the joint embedding
property. As a first approximation to this problem, we prove the undecidability
of the corresponding decision problem in the category of graphs. Modifying this
proof also gives the undecidability, in the category of graphs, of the
corresponding decision problem for the joint homomorphism property, which is of
interest in infinite-domain constraint satisfaction problems.
