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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 finda continuous transition for $N\geq M+4$. Estimates of the critical exponentsfor 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 ingood agreement with the results of the non-perturbative renormalization groupapproach. For $M\geq2$ we find estimates of the exponents and marginaldimensionality between the values obtained in the first and second orders ofthe large-N expansion. To complete the picture, we also consider the usual$O(N)$ model ($M=1$).

Phase transition in the three-dimensional $O(N)\otimes O(M)$ model: a Monte Carlo study