We study the thermodynamical properties and lattice dynamics of two-dimensional crystalline membranes, such as graphene and related compounds, in zero temperature limit, where quantum effects are dominant. We find out that, just as in the high temperature classical limit, a fundamental role is played by the anharmonic coupling between in-plane and out-of plane lattice modes, which leads to a strong reconstruction of the dispersion relation of the out-of-plane mode. We identify a crossover temperature, T
*, bellow which quantum effects are dominant. We estimate that for graphene T
* ∼ 70 - 90 K. Inclusion of anharmonic effects makes the thermal expansion finite in the thermodynamic limit, and below T
* it tends to zero as a power law as T→0 as required by the third law of thermodynamics. The specific heat also goes to zero as a power law as T→0, but with a exponent that differs from the one predicted by the harmonic theory.