In this chapter we will be considering the effect of confining polygons to lie in a bounded geometry. This has already been briefly discussed in Chapters 2 and 3, but here we give many more results. The simplest, non-trivial case is that of SAP on the two-dimensional square lattice ℤ2, confined between two parallel lines, say x = 0 and x = w. This problem is essentially 1-dimensional, and as such is in principle solvable. As we shall show, the solution becomes increasingly unwieldy as the distance w between the parallel lines increases. Stepping up a dimension to the situation in which polygons in the simple-cubic lattice ℤ3 are confined between two parallel planes, that is essentially a two-dimensional problem, and as such is not amenable to exact solution. Self-avoiding walks in slits were first treated theoretically by Daoud and de Gennes  in 1977, and numerically by Wall et al.  the same year. Wall et al. studied SAW on ℤ2, in particular the mean-square end-to-end distance. For a slit of width one they obtained exact results, and also obtained asymptotic results for a slit of width two. Around the same time, Wall and co-workers [13, 15] used Monte Carlo methods to study the width dependence of the growth constant for walks confined to strips of width w. In 1980 Klein  calculated the behaviour of SAW and SAP confined to strips in ℤ2 of width up to six, based on a transfer matrix formulation.