This paper provides an overview of the Europa Jupiter System Mission (EJSM) and of its scientific objectives, focusing the attention on the subsurface radar (SSR) instrument included in the model payload of the Jupiter Ganymede Orbiter (JGO). The SSR instrument is a radar sounder system at low frequency (HF/VHF band) designed to penetrate the surface of Ganymede icy moon of Jupiter for performing a subsurface analysis with a relatively high range resolution. This active instrument is aimed at acquiring information on the Ganymede (and partially on the Callisto during flybys) shallow subsurface. The paper addresses the main issues related to SSR, presenting its scientific goals, describing the concept and the design procedure of the instrument, and illustrating the signal processing techniques. Despite the fact that SSR can be defined on the basis of the heritage of the Mars Advanced Radar for Subsurface and Ionosphere Sounding (MARSIS) and SHAllow RADar (SHARAD) instruments currently operating at Mars, the EJSM mission poses additional scientific and technical challenges for its design: 1) the presence of a relevant Jupiter radio emission (which is very critical because it has a significant power spectral density in proximity of the expected SSR central frequency); 2) the properties of the subsurface targets, which are different from those of the Mars subsurface; 3) the different orbit conditions; and 4) the limited available resources (in terms of mass, power, and downlink data rate). These challenges are analyzed and discussed in relation to the design of the instrument in terms of: 1) choice of the central frequency and the bandwidth; 2) signal-to-noise ratio (SNR); 3) signal-to-clutter ratio (SCR); and 4) definition of the synthetic aperture processing. Finally, the procedure defined for SSR performance assessment is described and illustrated with some numerical examples.