Ferroelectric solids display a host of interesting electrical, mechanical and optical properties, and are thus used in a variety of mechanical, electronic and photonic devices. These solids are spontaneously electrically polarized, and the polarization distribution in a material often form intricate patterns known as domain pattern. Further, the polarization and domain patterns can be affected by mechanical stress, electrical voltage and optical illumination, giving rise to interesting properties. This talk will describe a number of variational problems motivated by the study of ferroelectric solids. It will describe how models of ferroelectric materials lead to non-convex variational problems whose solutions contain fine-scale oscillations. It will also describe some asymptotic problems and how they can be addressed using Gamma-convergence. This talk will provide the necessary physical and mathematical background, illustrate the key ideas using simple examples, describe theoretical predictions and discuss their experimental validation.