In this thesis we explore the rich physics of defect formation and dynamics in both spherical crystals and spherical nematics. For spherical crystals we describe the dynamics of scar defects and provide an analytical determination of the elastic spring constants of dislocations within scars using continuum elasticity theory with defect sources. Our results are compared to experimental observations of spherical crystals created in Pickering emulsions. We also discuss the remarkable behavior of interstitials and vacancies in spherical crystals. Interstitials or vacancies in a sufficiently large spherical crystal are, unlike in flat space, unstable to fragmentation into several individual dislocations, each of which glides towards the nearest disclination, eventually forming grain boundary scars. Using numerical simulations of the generalized Thomson problem we investigate interstitial fractionalization in some detail. We determine the dependence of final state energies on initial conditions and compare the position dependence of interstitial energies with the predictions of continuum elasticity theory on the sphere. For spherical nematics we combine Monte Carlo simulations and analytical calculations to explore the organization of defects in a system of hard and soft rods strictly confined to the two-dimensional sphere. In the ground state, orientationally ordered nematic phases with four type +1/2 disclinations are clearly observed. The disclinations are found to lie on a great circle rather than at the vertices of a tetrahedron. We explain this phenomenon in terms of the Frank free energy for an anisotropic spherical nematic in which splay deformations are far softer than bend deformations. We further examine the influence of the anisotropy of splay and bend elastic constants on the relative location of defects by minimizing the discretized strain energy from a mesoscopic Landau-de Gennes tensor model. From this we demonstrate that defect positions can be controlled by varying the anisotropy of the splay and bend moduli. Such control may prove useful in designing mesoscopic molecules and bulk materials by attaching ligands to functionalize the defect sites.