  We study large random dissections of polygons. We consider random dissections
of a regular polygon with $n$ sides, which are chosen according to Boltzmann
weights in the domain of attraction of a stable law of index $\theta\in(1,2]$.
As $n$ goes to infinity, we prove that these random dissections converge in
distribution toward a random compact set, called the random stable lamination.
If $\theta=2$, we recover Aldous' Brownian triangulation. However, if
$\theta\in(1,2)$, large faces remain in the limit and a different random
compact set appears. We show that the random stable lamination can be coded by
the continuous-time height function associated to the normalized excursion of a
strictly stable spectrally positive L\'{e}vy process of index $\theta$. Using
this coding, we establish that the Hausdorff dimension of the stable random
lamination is almost surely $2-1/\theta$.
