  We study the $O(N)$-invariant $\phi^4$ model on the simple cubic lattice by
using Monte Carlo simulations. By using a finite size scaling analysis, we
obtain accurate estimates for the critical exponents $\nu$ and $\eta$ for
$N=4$, $5$, $6$, $8$, $10$, and $12$. We study the model for each $N$ for at
least three different values of the parameter $\lambda$ to control leading
corrections to scaling. We compare our results with those obtained by other
theoretical methods.
