  We study the finite element approximation of the Kirchhoff plate equation on
domains with curved boundaries using the Hellan-Herrmann-Johnson (HHJ) method.
We prove optimal convergence on domains with piecewise $C^{k+1}$ boundary for
$k \geq 1$ when using a parametric (curved) HHJ space. Computational results
are given that demonstrate optimal convergence and how convergence degrades
when curved triangles of insufficient polynomial degree are used. Moreover, we
show that the lowest order HHJ method on a polygonal approximation of the disk
does not succumb to the classic Babu\v{s}ka paradox, highlighting the
geometrically non-conforming aspect of the HHJ method.
