  We study the wetting model, which considers a random walk constrained to
remain above a hard wall, but with additional pinning potential for each
contact with the wall. This model is known to exhibit a wetting phase
transition, from a localized phase (with trajectories pinned to the wall) to a
delocalized phase (with unpinned trajectories). As a preamble, we take the
opportunity to present an overview of the model, collecting and complementing
well-known and other folklore results. Then, we investigate a version with
elevated boundary conditions, which has been studied in various contexts both
in the physics and the mathematics literature; it can alternatively be seen as
a wetting model in a square well. We complement here existing results, focusing
on the equilibrium properties of the model, for a general underlying random
walk (in the domain of attraction of a stable law). First, we compute the free
energy and give some properties of the phase diagram; interestingly, we find
that, in addition to the wetting transition, a so-called saturation phase
transition may occur. Then, in the so-called Cram\'er's region, we find an
exact asymptotic equivalent of the partition function, together with a (local)
central limit theorem for the fluctuations of the left-most and right-most
pinned points, jointly with the number of contacts at the bottom of the well.
