Liquid crystals (LCs), presently the basis of the dominant electronics display technology, also hold immense potential for the design of new self-assembling, self-healing, and "smart" responsive materials. Essential to many of these novel materials are liquid crystalline defects, places where the liquid crystalline order is forced to break down, replacing the LC locally with a higher-symmetry phase. Despite the energetic cost of this local melting, defects are often present at equilibrium when boundary conditions frustrate the material order. These defects provide micron-scale tools for organizing colloids, focusing light, and generating micropatterned materials. Manipulating the shapes of the boundaries thus offers a route to obtaining new and desirable self-assembly outcomes in LCs, but each added degree of complexity in the boundary geometry increases the complexity of the liquid crystal's response. Therefore, conceptually minimal changes to boundary geometry are investigated for their effects on the self-assembled defect arrangements that result in nematic and smectic-A LCs in three dimensions as well as two-dimensional smectic LCs on curved substrates. In nematic LCs, disclination loops are studied in micropost confining environments and in the presence of sharp-edged colloidal inclusions, using both numerical modeling and topological reasoning. In both scenarios, sharp edges add new possibilities for the shape or placement of disclinations, permitting new types of colloidal self-assembly beyond simple chains and hexagonal lattices. Two-dimensional smectic LCs on curved substrates are examined in the special cases where the substrate curvature is confined to points or curves, providing an analytically tractable route to demonstrate how Gaussian curvature is associated with disclinations and grain boundaries, as well as these defects' likely experimental manifestations. In three-dimensional smectic-A LCs, novel self-assembled arrangements of focal conic domains (FCDs) are shown to arise from geometric patterning or curvature in boundaries exhibiting so-called hybrid anchoring. These new arrangements allow control over both the packing of the FCDs and their eccentricities. In general, defect self-assembly behavior in LCs is shown to depend sensitively on the shapes of confining boundaries, colloidal inclusions, and substrates, and several broad, new geometrical principles for directing the assembly of nontrivial defect configurations are presented.