White-dwarf asteroseismology with the Kepler space telescope

In the course of their evolution, white-dwarf stars go through at least one phase of variability in which the global pulsations they undergo allow astronomers to peer into their interiors, this way making possible to shed light on their deep inner structure and evolutionary stage by means of asteroseismology. The study of pulsating white dwarfs has witnessed substantial progress in the last decade, and this has been so largely thanks to the arrival of continuous observations of unprecedented quality from space, like those of the CoRoT, Kepler, and TESS missions. This, along with the advent of new detailed thoretical models and the development of improved asteroseismological techniques, has helped to unravel the internal chemical structure of many pulsating white dwarfs, and, at the same time, has opened new questions that challenge theoreticians. In particular, uninterrupted monitoring of white-dwarf stars for months has allowed discovering phenomena impossible to detect with ground-based observations, despite admirable previous efforts like the Whole Earth Telescope (WET). Here, we start by reviewing the essential properties of white-dwarf and pre-white dwarf stars and their pulsations, and then, we go through the different families of pulsating objects known to date. Finally, we review the most outstanding findings about pulsating white dwarfs and pre-white dwarfs made possible with the unprecedented-quality observations of the Kepler space telescope, although we envisage that future analyzes of space data from this mission that still await to be examined may reveal new secrets of these extremely interesting variable stars.