  We report our density matrix renormalization-group and analytical work on S=1
antiferromagnetic Heisenberg spin chains. We study the finite size behavior
within the framework of the non-linear sigma model. We study the effect of
magnon-magnon interactions on the finite size spectrum and on the magnetization
curve close to the critical magnetic field, determine the magnon scattering
length and compare it to the prediction from the non-linear $\sigma$ model.
