Over the last several years, we have developed a comprehensive program aimed at better understanding the properties of pulsating DA white dwarfs (or ZZ Ceti stars). These stars are nonradial pulsators of the g-type, and their study can lead to inferences about their internal structure. For instance, the period spectrum of a white dwarf is most sensitive to its vertical chemical stratification, and one of the major goals of white dwarf seismology is to determine the thickness of the hydrogen layer that sits on top of a star. This can be done, in principle, by comparing in detail theoretical period spectra with the periods of the observed excited modes. Likewise, because the cooling rate of a white dwarf is very sensitive to the specific heat of its core material (and hence to its composition), it is possible to infer the core composition through measurements and interpretations of rates of period change in a pulsator.