  We compute the beta-function and the anomalous dimension of all the
non-derivative operators of the theory up to three-loops for the most general
nearest-neighbour O(N)-invariant action together with some contributions to the
four-loop beta-function. These results are used to compute the first analytic
corrections to various long-distance quantities as the correlation length and
the general spin-$n$ susceptibility. It is found that these corrections are
extremely large for $RP^{N-1}$ models (especially for small values of N), so
that asymptotic scaling can be observed in these models only at very large
values of beta. We also give the first three terms in the asymptotic expansion
of the vector and tensor energies.
