The first European mission dedicated to the exploration of Jupiter and its icy moons (JUpiter ICy moons Explorer - JUICE) will be launched in 2022 and will reach its final destination in 2030. The main goals of this mission are to understand the internal structure of the icy crusts of three Galilean satellites (Europa, Ganymede and Callisto) and, ultimately, to detect Europa's subsurface ocean, which is believed to be the closest to the surface amongst those hypothesized to exist on these moons. JUICE will be equipped with the 9 MHz subsurface penetrating radar RIME (Radar for Icy Moon Exploration), which is designed to image the ice down to a depth of 9 km. A parallel mission, Europa Clipper, equipped with 9 MHz and 60 MHz radar antennas, is also currently under study by NASA. The success of these experiments strongly relies on the accurate prediction of the radar performance and on the optimal processing and interpretation of radar echoes that, in turn, depend on the dielectric properties of the materials composing the icy satellite crusts. In the present review we report a complete range of potential ice types that may occur on these icy satellites to understand how they may affect the results of the proposed missions. First we discuss the experimental results on pure and doped water ice in the framework of the Jaccard theory, highlighting the critical aspects in terms of a lack of standard laboratory procedures and inconsistency in data interpretation. We then describe the dielectric behavior of extra-terrestrial ice analogs like hydrates and icy mixtures, carbon dioxide ice and ammonia ice. Building on this review we have selected the most suitable data to compute dielectric attenuation, velocity, vertical resolution, and reflection coefficients for such icy moon environments, with the final goal being to estimate the potential capabilities of the radar missions as a function of the frequency and temperature ranges of interest for the subsurface sounders. We present the different subsurface scenarios and associated radar signal attenuation models that have been proposed so far to simulate the structure of the crust of Europa, and discuss the physical and geological nature of various dielectric targets potentially detectable with RIME. Finally, we briefly highlight several unresolved issues that should be addressed, in near future, to improve our capability to produce realistic electromagnetic models of icy moon crusts. The present review is of interest for the geophysical exploration of all solar system bodies, including the Earth, where ice can be present at the surface or at relatively shallow depths.