  We investigate the low-temperature behavior of two-dimensional (2D)
RP$^{N-1}$ models, characterized by a global O($N$) symmetry and a local
${\mathbb Z}_2$ symmetry. For $N=3$ we perform large-scale simulations of four
different 2D lattice models: two standard lattice models and two different
constrained models. We also consider a constrained mixed O(3)-RP$^2$ model for
values of the parameters such that vector correlations are always disordered.
We find that all these models show the same finite-size scaling (FSS) behavior,
and therefore belong to the same universality class. However, these FSS curves
differ from those computed in the 2D O(3) $\sigma$ model, suggesting the
existence of a distinct 2D RP$^2$ universality class. We also performed
simulations for $N=4$, and the corresponding FSS results also support the
existence of an RP$^3$ universality class, different from the O(4) one.
