  We present and summarize the results of recent studies on non-Euclidean
plates with imposed constant negative Gaussian curvature in both the F\"oppl -
von K\'arm\'an and Kirchhoff approximations. Motivated by experimental results
we focus on annuli with a periodic profile. We show that in the F\"oppl - von
K\'arm\'an approximation there are only two types of global minimizers -- flat
and saddle shaped deformations with localized regions of stretching near the
boundary of the annulus. We also show that there exists local minimizers with
$n$-waves that have regions of stretching near their lines of inflection. In
the Kirchhoff approximation we show that there exist exact isometric immersions
with periodic profiles. The number of waves in these configurations is set by
the condition that the bending energy remains finite and grows approximately
exponentially with the radius of the annulus. For large radii, these shape are
energetically favorable over saddle shapes and could explain why wavy shapes
are selected by crocheted models of the hyperbolic plane.
