Exact form factors of the O(N) $\sigma$-model

A general form factor formula for the $O(N)\sigma$-model is constructed and applied to several operators. The large N limits of these form factors are computed and compared with the 1/N expansion of the $O(N)\sigma$-model in terms of Feynman graphs and full agreement is found. In particular, O(3) and O(4) form factors are discussed. For the $O(3)\sigma$-model several low particle form factors are calculated explicitly.