Giant dielectric constant materials have great potential in application of miniaturization of electronic devices and high density energy storage. In the present work, the structure and giant dielectric response have been systematically investigated in 0.4Li<,0.05>Ni<,0.95>O/0.6SrTiO<,3> composite ceramics and LuFeO<,3> ceramics. The following primary results and conclusions have been obtained. Novel high dielectric constant composite materials were prepared by incorporating the dispersed Li<,0.05>Ni<,0.95> particles into SrTiO<,3> matrix. A dielectric constant in the order of 104 was obtained in the composite with 40mo1％ Li<,0.05>Ni<,0.95>. In 0.4Li<,0.05>Ni<,0.95>O/0.6SrTiO<,3> composites, the low-temperature dielectric relaxation ascribed to the electron hoping between Ti<3+> and Ti<4+> because of the processing of thermal activation. The medi-temperature dielectric relaxations is significantly suppressed by the O<,2>-annealing, while the low and high-temperature dielectric relaxation is just affected slightly. Since there is no crystal structure change, the medi-temperature dielectric relaxations is deeply related with the oxygen vacancy. There a significant increasing for the dielectric constant in the high-temperature, as we know, the separated conducting particles approach to each other, which lead to the thinner of the small capacitors between two semi-conducting particles, and hence the higher effective dielectric constant of the composites. Hetero-structures formed in the interfaces of Li<,0.05>Ni<,0.95>O and SrTiO<,3> during the sintering progress, which hindered the movements of oxygen vacancies near the interfaces, leading to a dielectric relaxation at high temperatures. The dielectric properties of dense LuFeO3 ceramics prepared by solid state sintering process have also been characterized. An giant dielectric constant step is observed in LuFeO<,3> ceramics, and there is a frequency-dependent critical temperature where the dielectric constant drops quickly when the sample is cooled down through there. A very high relaxor-like dielectric peak with strong frequency dispersion is also observed in the higher temperature range. The high-temperature dielectric relaxations is significantly suppressed by the O2-annealing, while the low-temperature dielectric relaxation is just affected slightly. As the results, the dielectric constant peak is reduced significantly, and the dielectric constant step is extended and the magnitude is decreased. Since there is no crystal structure change, the only possible structuralchange is the reduced oxygen vacancy concentration and the subsequent reduction of Fe<2+> content to keep the electric compensation. These results confirm the relaxation peaks are observed on the curve of dielectric loss vs temperature. According to analysis of XPS, the estimated ratio of Fe<2+>: Fe<3+> is about 1:2. The presence of Fe2~ ions in the LuFeO<,3> ceramics is compensated by the presence of oxygen vacancies. The mixed-valance structure of Fe<2+>/Fe<3+> offers the possibility of charge ordering, and subsequently offers the structure origin for the low-temperature relaxor-like dielectric behavior and the electronic ferroelectricity. The high-temperature dielectric relaxation is deeply related to the oxygen vacancies.