We discuss the asymptotic behaviour of models of lattice polygons, mainly on
the square lattice. In particular, we focus on limiting area laws in the
uniform perimeter ensemble where, for fixed perimeter, each polygon of a given
area occurs with the same probability. We relate limit distributions to the
scaling behaviour of the associated perimeter and area generating functions,
thereby providing a geometric interpretation of scaling functions. To a major
extent, this article is a pedagogic review of known results.