  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.
