  We argue that the two-dimensional $O(N)$-invariant lattice $\sigma$-model
with mixed isovector/isotensor action has a one-parameter family of nontrivial
continuum limits, only one of which is the continuum $\sigma$-model constructed
by conventional perturbation theory. We test the proposed scenario with a
high-precision Monte Carlo simulation for $N=3,4$ on lattices up to $512 \times
512$, using a Wolff-type embedding algorithm. [CPU time $\approx$ 7 years IBM
RS-6000/320H] The finite-size-scaling data confirm the existence of the
predicted new family of continuum limits. In particular, the $RP^{N-1}$ and
$N$-vector models do not lie in the same universality class.
