  Using the renormalization group approach, we consider the $O(N)\otimes O(M)$
model in four and more dimensions. We find that independently on $N$ and $M$,
for $N\geq M\geq 2$, a transition can be of both the first and second order. In
$d>4$, we also cannot exclude a pseudo-first-order behavior. As specific
physically interesting cases, we consider the lattice version of the
$O(2)\otimes O(2)$, $O(3)\otimes O(2)$ and $O(3)\otimes O(3)$ sigma models on a
four dimensional hypercubic lattice. In all these cases, we find a distinct
first-order transition.
