We present an overview of current state-of-the art measurements and knowledge of dielectric properties of ice and snow. They are the fundamental property for electromagnetic applications in cryospheric sciences, like ground-penetrating radar (GPR) and satellite remote sensing. Relevance ranges from improved determination of snow-cover properties for improved avalanche risk estimation to reconstruction of past climate signals from ice cores to improved understanding of dynamics of ice masses like Greenland and Antarctica. Our results are based on several techniques: (i) laboratory observations with a coaxial cell (CC) set-up, employed to measure dielectric properties on artificial and natural snow and ice samples in the range from 1 MHz to 1.5 GHz; (ii) dielectric profiling (DEP) of firn and ice cores in the range of 100 kHz; and (iii) usage of GPR to indirectly deduce the dielectric properties of the bulk medium via variations in wave speed and reflection coefficients. In contrast to many other substances, H2O as the underlying molecule, exists close to its melting point under ordinary conditions, involving snow and ice on Earth. This is of large interest for environmental applications as the so-called snow-water equivalent (i.e. the total mass) can greatly vary depending on density and liquid-water saturation of a snow cover, without showing considerable changes in snow height. In contrast, it poses a major problem for determining dielectric properties close to the melting point, especially in the laboratory, as the medium of interest partly undergoes a phase transition while varying temperature. In addition, especially for applications on glaciers and ice sheets, the anisotropic nature of ice has to be taken into account, as ice viscosity - and thus flow behavior - varies over four orders of magnitude depending on the crystal orientation fabric.