This thesis consists of four independent papers. In the first paper, joint with Kechris, we study the global aspects of structurability in the theory of countable Borel equivalence relations. For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We show that K-structurability interacts well with various preorders commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K; in particular, we characterize the K such that every K-structurable equivalence relation is smooth. In the second paper, we consider the classes of Kn-structurable equivalence relations, where Kn is the class of n-dimensional contractible simplicial complexes. We show that every Kn-structurable equivalence relation Borel embeds into one structurable by complexes in Kn with the further property that each vertex belongs to at most Mn := 2n-1(n2+3n+2)-2 edges; this generalizes a result of Jackson-Kechris-Louveau in the case n=1. In the third paper, we consider the amalgamation property from model theory in an abstract categorical context. A category is said to have the amalgamation property if every pushout diagram has a cocone. We characterize the finitely generated categories I such that in every category with the amalgamation property, every I-shaped diagram has a cocone. In the fourth paper, we prove a strong conceptual completeness theorem (in the sense of Makkai) for the infinitary logic Lω1ω: every countable Lω1ω-theory can be canonically recovered from its standard Borel groupoid of countable models, up to a suitable syntactical notion of equivalence. This implies that given two theories (L,T) and (L',T'), every Borel functor Mod(L',T') → Mod(L,T) between the respective groupoids of countable models is Borel naturally isomorphic to the functor induced by some L'ω1ω-interpretation of T in T', which generalizes a recent result of Harrison-Trainor, Miller, and Montalban in the case where T, T' are ℵ0-categorical.