  Many extremely low-mass (ELM) white-dwarf (WD) stars are currently being
found in the field of the Milky Way. Some of these stars exhibit long-period
nonradial $g$-mode pulsations, and constitute the class of ELMV pulsating WDs.
In addition, several low-mass pre-WDs, which could be precursors of ELM WDs,
have been observed to show short-period photometric variations likely due to
nonradial $p$ modes and radial modes. They could constitute a new class of
pulsating low-mass pre-WD stars, the pre-ELMV stars. Here, we present the
recent results of a thorough theoretical study of the nonadiabatic pulsation
properties of low-mass He-core WDs and pre-WDs on the basis of fully
evolutionary models representative of these stars.
