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Many extremely low-mass (ELM) white-dwarf (WD) stars are currently beingfound in the field of the Milky Way. Some of these stars exhibit long-periodnonradial $g$-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 tononradial $p$ modes and radial modes. They could constitute a new class ofpulsating low-mass pre-WD stars, the pre-ELMV stars. Here, we present therecent results of a thorough theoretical study of the nonadiabatic pulsationproperties of low-mass He-core WDs and pre-WDs on the basis of fullyevolutionary models representative of these stars.

Recent advances in the theoretical modeling of pulsating low-mass He-core white dwarfs