  Analytical and numerical methods are applied to principal chiral models on a
two-dimensional lattice and their predictions are tested and compared. New
techniques for the strong coupling expansion of SU(N) models are developed and
applied to the evaluation of the two-point correlation function. The
momentum-space lattice propagator is constructed with precision O(\beta^{10})
and an evaluation of the correlation length is obtained for several different
definitions. Three-loop weak coupling contributions to the internal energy and
to the lattice $\beta$ and $\gamma$ functions are evaluated for all N, and the
effect of adopting the ``energy'' definition of temperature is computed with
the same precision. Renormalization-group improved predictions for the
two-point Green's function in the weak coupling ( continuum ) regime are
obtained and successfully compared with Monte Carlo data. We find that strong
coupling is predictive up to a point where asymptotic scaling in the energy
scheme is observed. Continuum physics is insensitive to the effects of the
large N phase transition occurring in the lattice model. Universality in N is
already well established for $N \ge 10$ and the large N physics is well
described by a ``hadronization'' picture.
