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This paper describes the mission concepts, design, and achievements of the Italian Space Agency (ASI)-provided Mars SHAllow RADar (SHARAD) sounder high-frequency (HF) sounding radar, used onboard the National Aeronautics and Space Administration (NASA) Mars Reconnaissance Orbiter (MRO) Spacecraft. Its goals are the detection of liquid or solid water below the surface, and the mapping of subsurface geologic structures. Following a brief overview of the MRO mission and of its main science objectives, the paper introduces the basic principles of operation of the radar sounder, and addresses the major design issues faced by such a system. The greatest challenges faced in the design are the control of the interference from off-nadir echoes and the need for a high signal fidelity over a very large fractional bandwidth. The core of the paper is devoted to describing how the above problems have been tackled in the design of the SHARAD instrument, and the main characteristics of its architecture. The two key features of the instrument system design are 1) generation of the transmitted signal directly at the transmitted frequency; and 2) sampling performed directly at the radio frequency (by means of a subsampling technique). The careful design of these features, intended to keep the analog signal path very simple, minimizes distortions and stability problems. An overview of the calibration approach of both the system impulse response and the antenna gain at nadir versus solar array position, an assessment of the in-flight performance of the instrument, and a short summary of the achieved science results are also provided.