  We demonstrate that a key elastic parameter of a suspended crystalline
membrane---the Poisson ratio (PR) $\nu$---is a non-trivial function of the
applied stress $\sigma$ and of the system size $L$, i.e., $\nu=\nu_L(\sigma)$.
We consider a generic 2D membrane embedded into space of dimensionality
$2+d_c$. (The physical situation corresponds to $d_c=1$.) A particularly
important application of our results is free-standing graphene. We find that at
very low stress, where the membrane exhibits a linear response, the PR
$\nu_L(0)$ decreases with increasing $L$ and saturates for $ L\to \infty$ at a
value which depends on the boundary conditions and is essentially different
from the value $\nu=-1/3$ previously predicted by the membrane theory within a
self-consisted scaling analysis. By increasing $\sigma$, one drives a membrane
into a non-linear regime characterized by a universal value of PR that depends
solely on $d_c.$ This universal non-linear PR acquires its minimum value
$\nu_{\rm min}=-1$ in the limit $d_c\to \infty.$ With the further increase of
$\sigma$, the PR changes sign and finally saturates at a positive non-universal
value prescribed by the conventional elasticity theory. We also show that one
should distinguish between the absolute and differential PR ($\nu$ and
$\nu^{\rm diff}$, respectively). While coinciding in the limits of very low and
very high stresses, they differ in general, $\nu \neq \nu^{\rm diff}$. In
particular, in the non-linear universal regime, $\nu^{\rm diff}$ takes a
universal value which, similarly to absolute PR, is a function solely of $d_c$
but is different from the universal value of $\nu$. In the limit $d_c\to
\infty$, the universal value of $\nu^{\rm diff}$ tends to $-1/3$, at variance
with the limiting value $-1$ of $\nu$. Finally, we briefly discuss
generalization of these results to a disordered membrane.
