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Lattice $SU(N)\times SU(N)$ chiral models are analyzed by strong and weakcoupling expansions and by numerical simulations. $12^{th}$ order strongcoupling series for the free and internal energy are obtained for all $N\geq6$. Three loop contributions to the internal energy and to the lattice$\beta$-function are evaluated for all $N$ and non-universal corrections to theasymptotic $\Lambda$ parameter are computed in the ``temperature'' and the``energy'' scheme. Numerical simulations confirm a faster approach toasymptopia of the energy scheme. A phenomenological correlation between thepeak in the specific heat and the dip of the $\beta$-function is observed.Tests of scaling are performed for various physical quantities, findingsubstantial scaling at $\xi \gtrsim 2$. In particular, at $N=6$ three differentmass ratios are determined numerically and found in agreement, withinstatistical errors of about 1\%, with the theoretical predictions from theexact S-matrix theory.

Two dimensional SU(N) x SU(N) chiral models on the lattice