Thin elastic sheets are very common in both natural and man-made structures. The configurations these structures assume in space are often very complex and may contain many length scales, even in the case of unconstrained thin sheets. We will show that in some cases, a simple intrinsic geometry leads to complex three-dimensional configurations, and discuss the mechanism shaping thin elastic sheets through the prescription of an intrinsic metric. Current reduced (two-dimensional) elastic theories devised to describe thin structures treat either plates (flat bodies having no structure along their thin dimension) or shells (non-flat bodies having a non-trivial structure along their thin dimension). We propose the concept of non-Euclidean plates, which are neither plates nor shells, to approximate many naturally formed thin elastic structures. We derive a thin plate theory which is a generalization of existing linear plate theories for large displacements but small strains, and arbitrary intrinsic geometry. We conclude by surveying some experimental results for laboratory-engineered non-Euclidean plates.