  We consider here the percolation problem in thin films, both in the direction
normal to the film and in the direction parallel to the film. We thereby
describe here the cross-over between 2D and 3D percolation, which we do on
cubic and square lattices. The main relations are derived using scaling and
real space renormalisation arguments. They are checked by numerical
simulations, which also provide the numerical prefactors. We calculate in
particular the correlation length parallel to the film, the average mass and
the mass distribution $n(m)$ of the clusters. In particular, we show that the
latter is given by a master function of $h^{-D+1/\sigma_{2}\nu_{3}}|
p-p_{c}(h)|^{1/\sigma_{2}} m$, where $h$ is the thickness of the film and
$D,\nu_3,\sigma_2$ are tabulated 2D and 3D critical exponents. $p_c(h)$ is the
percolation threshold of the film which we also calculate. These results are of
interest in particular for describing the glass transition in thin polymer
films.
