The self-avoiding walk is a random walk that is not allowed to
visit the same site more than once. The distance from the starting
point to the end point is a random variable whose variance is
believed to scale like the number of steps in the walk raised to
the 3/2 power in two dimensions. We present the results of
Monte Carlo simulations of this model on several different
two-dimensional lattices. They show that the distribution of this
walk does not depend on the lattice in the limit that the number of
steps goes to infinity.