The trapping and interaction of nano- and micro-particles at fluid interfaces is broadly important in technology, including established technologies like froth flotation to recover ores and emulsion stabilization by adsorbed particle layers, or the formation of Pickering emulsions. Particle trapping and organization at interfaces is also important in advanced materials processing, for example, in the formation of ordered nanoparticle structures at fluid interfaces in Langmuir troughs. Capillary interactions are ubiquitous between microscale colloids trapped at fluid interfaces. These interactions are particularly interesting because they depend on the shape of the interface shape and the shape of the particle’s contact line. This latter quantity depends on the particles’ shape, its surface chemistry and topography, but not on its bulk materials properties. Thus, these interaction are “materials agnostic”, and can be used to organize finely divided materials of diverse bulk properties. Generally, colloids become trapped at fluid interfaces because upon adsorption, they eliminate a patch of liquid-liquid or liquid-vapor interface, significantly reducing the free energy of the system. Particles trapped at fluid interfaces generally have pinned, undulated contact lines that distort the interface around them. To minimize the area, and therefore the energy of these distortions, colloids interact and assemble. These interactions are significant at planar interfaces, and depend strongly on the shape of the host interface. In particular, on curved fluid interfaces, capillary interactions direct isolated colloid motion along paths defined by deviatoric curvature gradients. This directed motion relies on the leading order, long-ranged quadrupolar distortions made by the colloids' undulated pinned contact lines, and the underlying “saddle-like” shape of the interface that is also described by a quadrupolar term in the limit of small interfacial slopes. While the importance of curvature for isolated particles on curved interfaces is now well appreciated, its role in guiding structure formation has not been well studied. In my thesis, I study the organization of microparticles trapped on curved interfaces using theory and experiment. I first focus on pair interactions between particles with pinned contact lines trapped on interfaces with curvature gradients. I derive closed-form expressions for the interaction potential between spherical or disk-like particles with nearly circular contact lines. Gradients in this potential predict the forces on the particles and hence particle paths. The particles are attracted to each other, and are also attracted to zones of high deviatoric curvature. Depending on the relative magnitudes of their attraction and the strength of the local curvature gradient, particle pairs are predicted to dimerize, co-migrate without dimerization but in each other’s influence, or to migrate independently. On curved oil-water interfaces, I study pair interactions and dimer formation of spherical microparticles. These particles induce significant nanometric deformation on the interface. Particles are attracted to each other and come to apparent contact. I use the analytical pair potentials to understand dimerization and to identify criteria for dimers to form. When many particles are present on the curved interface, they organize in nearly trapped structures that reflect the underlying curvature field. To model this process, I use Monte Carlo simulations in which the energy landscape is determined by the analytical pair potential in the curved interface, extended to have higher order modes in the particle contact line undulation and in the local expansion of the interface. By randomly adding particles with given contact line undulations, the Monte Carlo simulations generate structures with qualitative features similar to those formed in experiment. Lastly, I address pair interactions of elongated particles on curved fluid interfaces. The leading order disturbance made by each particle in the interface is a quadrupole in elliptical coordinates. The analytical pair potential now captures the alignment of the particle’s major axis with respect to the underlying curvature field. I compare prediction to the paths taken by cylindrical microparticle pairs in a curvature field. The study of structure formation by Monte Carlo is left for future work.