  We prove that Kellendonk's $C^*$-algebra of an aperiodic and repetitive
tiling with finite local complexity is classifiable by the Elliott invariant.
Our result follows from showing that tiling $C^*$-algebras are
$\mathcal{Z}$-stable, and hence have finite nuclear dimension. To prove
$\mathcal{Z}$-stability, we extend Matui's notion of almost finiteness to the
setting of \'etale groupoid actions following the footsteps of Kerr. To use
some of Kerr's techniques we have developed a version of the Ornstein-Weiss
quasitiling theorem for general \'etale groupoids.
