DescriptionUsing a combination of numerical and analytical techniques, I present characterizations of ferroelectric materials in bulk, thin-film and nanostructure geometries. My results have impact on ongoing research and on design for nanodevices. Size-dependent effects in ferroelectrics are important due to their long-range electrostatic interactions; thus their dielectric properties depend on electromechanical boundary conditions. In my first study, I address the effects of strain on the measured properties of thin-film (TF) ferroelectrics. It has been suggested that the observed suppression of many TF dielectric characteristics implies underlying strain gradients in the film. I show that the same effects can be explained by a simpler model with homogeneous strain, and I suggest a “smoking gun” benchtop probe. The quantum paraelectric-ferroelectric transition (QPFT) is the topic of my second study. Using methods including finite-size scaling and self-consistent Gaussian theory, I calculate the classical-quantum crossover in the dielectric susceptbility and the resulting temperature-pressure phase diagram; comparison with current experiment is excellent and predictions are made for future measurements. Here, temperature can be considered a “finite-size effet” in time, and previous results on the QPFT using diagrammatic techniques are recovered and extended using this approach. Recent synthesis of artificially structured oxides with “checkerboard” patterning at the nanoscale has been reported, and this serves as motivation for my third study. Here, I use first-principles methods to characterize an atomic-scale BiFeO3-BiMnO3 nanocheckerboard, and find that it has properties that are distinctive from those of either parent compound. More specifically, it has both a spontaneous polarization and magnetization, and also displays a magnetostructural effect. My work on this prototypical multiferroic nanocheckerboard motivates further theoretical and experimental studies of new heterostructures with properties that are geometrically induced.