  Non-Euclidean plates are thin elastic bodies having no stress-free
configuration, hence exhibiting residual stresses in the absence of external
constraints. These bodies are endowed with a three-dimensional reference
metric, which may not necessarily be immersible in physical space. Here, based
on a recently developed theory for such bodies, we characterize the transition
from flat to buckled equilibrium configurations at a critical value of the
plate thickness. Depending of the reference metric, the buckling transition may
be either continuous or discontinuous. In the infinitely thin plate limit,
under the assumption that a limiting configuration exists, we show that the
limit is a configuration that minimizes the bending content, amongst all
configurations with zero stretching content (isometric immersions of the
mid-surface). For small but finite plate thickness we show the formation of a
boundary layer, whose size scales with the square root of the plate thickness,
and whose shape is determined by a balance between stretching and bending
energies.
