  Recent years have witnessed the discovery of many low-mass ($\lesssim 0.45
M_{\odot}$) white dwarf (WD) stars --- expected to harbor He cores--- in the
field of the Milky Way and in several galactic globular and open clusters.
Recently, three pulsating objects of this kind have been discovered: SDSS
J1840+6423, SDSS J1112+1117, and SDSS J1518+0658. Motivated by these very
exciting findings, and in view of the valuable asteroseismological potential of
these objects, we present here the main outcomes of a detailed theoretical
study on the seismic properties of low-mass He-core WDs based on fully
evolutionary models representative of these objects. This study is aimed to
provide a theoretical basis from which to interpret present and future
observations of variable low-mass WDs.
