  Using Monte Carlo simulations, we consider the lattice version of the
$O(N)\otimes O(M)$ sigma model for $2\leq M\leq4$ and $M\leq N \leq8$. We find
a continuous transition for $N\geq M+4$. Estimates of the critical exponents
for cases of second-order and weak first-order transitions are found. For $M=2$
our estimates of the exponents and marginal dimensionality $N_c^+(M)$ are in
good agreement with the results of the non-perturbative renormalization group
approach. For $M\geq2$ we find estimates of the exponents and marginal
dimensionality between the values obtained in the first and second orders of
the large-N expansion. To complete the picture, we also consider the usual
$O(N)$ model ($M=1$).
