A giant permittivity associated with the motion of domain walls is reported in epitaxial hetero- structures having alternating layers of ferroelectric and nonferroelectric oxides. At low frequencies, permittivities as high as 420 000 are found. Real and imaginary parts of the dielectric constant show large dispersion at high frequencies. In dc measurements, a nonlinear resistance is observed with a well-defined threshold field correlated with the dc bias-field dependence of ac permittivities. We interpret the observations as a result of the motion of a pinned domain wall lattice at low electric fields and sliding-mode motion at high electric fields.