Measurements for the determination of the complex permittivity of liquid and solid materials based on the use of whispering gallery (WG) dielectric resonators are presented. The procedure implies the measurement of the electromagnetic parameters of the involved resonator interacting with the material under study. The field distribution in the different WG resonant modes was obtained by an analytical calculation under the mode matching method approximation. The results of this calculation, combined with the experimental data, well account for the values of the complex permittivity of materials of interest. Due to the peculiar properties of WG resonators, the method shares the advantages of the wideband systems with the sensitivity and accuracy of the resonator systems in dielectric measurements. Dielectric permittivity of alumina (Al2O3) and of cyclohexane is measured in the frequency range 18-26 GHz. The overall accuracy of the measurements is discussed and the different sources of errors analyzed. Specific attention is paid to the measurements of variations of dielectric properties to monitor some physico-chemical processes. Finally, preliminary measurements at hundreds of GHz show that the procedure is particularly useful for study of dielectric properties up to the THz frequency region.