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The Radio & Plasma Wave Investigation (RPWI) onboard the ESA JUpiter ICy moons Explorer (JUICE) is described in detail. The RPWI provides an elaborate set of state-of-the-art electromagnetic fields and cold plasma instrumentation, including active sounding with the mutual impedance and Langmuir probe sweep techniques, where several different types of sensors will sample the thermal plasma properties, including electron and ion densities, electron temperature, plasma drift speed, the near DC electric fields, and electric and magnetic signals from various types of phenomena, e.g., radio and plasma waves, electrostatic acceleration structures, induction fields etc. A full wave vector, waveform, polarization, and Poynting flux determination will be achieved. RPWI will enable characterization of the Jovian radio emissions (including goniopolarimetry) up to 45 MHz, has the capability to carry out passive radio sounding of the ionospheric densities of icy moons and employ passive sub-surface radar measurements of the icy crust of these moons. RPWI can also detect micrometeorite impacts, estimate dust charging, monitor the spacecraft potential as well as the integrated EUV flux. The sensors consist of four 10 cm diameter Langmuir probes each mounted on the tip of 3 m long booms, a triaxial search coil magnetometer and a triaxial radio antenna system both mounted on the 10.6 m long MAG boom, each with radiation resistant pre-amplifiers near the sensors. There are three receiver boards, two Digital Processing Units (DPU) and two Low Voltage Power Supply (LVPS) boards in a box within a radiation vault at the centre of the JUICE spacecraft. Together, the integrated RPWI system can carry out an ambitious planetary science investigation in and around the Galilean icy moons and the Jovian space environment. Some of the most important science objectives and instrument capabilities are described here. RPWI focuses, apart from cold plasma studies, on the understanding of how, through electrodynamic and electromagnetic coupling, the momentum and energy transfer occur with the icy Galilean moons, their surfaces and salty conductive sub-surface oceans. The RPWI instrument is planned to be operational during most of the JUICE mission, during the cruise phase, in the Jovian magnetosphere, during the icy moon flybys, and in particular Ganymede orbit, and may deliver data from the near surface during the final crash orbit.