Compact stars that result from extreme mass loss on the asymptotic giant branch and planetary nebula formation are observed to pulsate in a very large surface effective temperature range as they cool to become the classical white dwarfs. The hottest and most luminous of these display periods in excess of 1000 seconds because they are large, but when the stars arrive on the cooling line on the Hertzsprung-Russell diagram, their periods become generally less than 1000 seconds. Then the stars have masses near 0.6 M⊙ and radii near 109 cm. Their luminosity depends then almost entirely on the surface effective temperature as the entire star with its legacy of complicated internal luminosity peaks cools to the classical simple electron degenerate structure. Very thin surface layers of hydrogen and helium cover the bulk of the carbon- and oxygen-rich mass that results from hydrogen and helium burning in earlier intermediate mass stellar evolution. The cause of the nonradial pulsations of low angular degree, but rather high radial order, for the most luminous of these stars is the cyclical ionization of carbon and oxygen in layers not too deep that their effectiveness is limited by a long luminosity time scale. Thus the surface hydrogen and helium must be thin, probably thinner than the current period spacings interpretation for PG 1159-035 suggests. For the classical DBV and DAV pulsators, it appears that neither the hydrogen ionization K and γ effects or convection blocking at the bottom of a hydrogen convection zone can destabilize the observed pulsations when the overriding short time scale effects of time-dependent convection are included. It appears, however, that a thin CO convection shell can produce pulsations by its time-dependent effects, but again only very thin H and He surface layers are allowed. This new pulsation mechanism can alleviate the serious problem that DAV variables are observed hotter than the hydrogen K and γ effects and convection blocking can predict. The appearance of non-pulsators in the DAV and DBV instability strips can be explained by a too-thick hydrogen and helium surface layer that interferes with (poisons) the CO ionization convection zone. Finally time-dependent convection predicts that only a few of the many possible modes exist due to their internal amplitude structure that can result in both strong driving and strong damping. Thus actually observed pulsating modes can assist in mapping individual internal white dwarf composition structures, not only by their periods but also the fact that they pulsate.