  Single-layer graphene is so flexible that its flexural mode (also called the
ZA mode, bending mode, or out-of-plane transverse acoustic mode) is important
for its thermal and mechanical properties. Accordingly, this review focuses on
exploring the relationship between the flexural mode and thermal and mechanical
properties of graphene. We first survey the lattice dynamic properties of the
flexural mode, where the rigid translational and rotational invariances play a
crucial role. After that, we outline contributions from the flexural mode in
four different physical properties or phenomena of graphene -- its thermal
conductivity, thermal expansion, Young's modulus, and nanomechanical resonance.
We explain how graphene's superior thermal conductivity is mainly due to its
three acoustic phonon modes at room temperature, including the flexural mode.
Its coefficient of thermal expansion is negative in a wide temperature range
resulting from the particular vibration morphology of the flexural mode. We
then describe how the Young's modulus of graphene can be extracted from its
thermal fluctuations, which are dominated by the flexural mode. Finally, we
discuss the effects of the flexural mode on graphene nanomechanical resonators,
while also discussing how the essential properties of the resonators, including
mass sensitivity and quality factor, can be enhanced.
