The primary scientific objective of the High Energy Solar Spectroscopic Imager (HESSI) is to understand particle acceleration and explosive energy release in the magnetized plasmas at the Sun. HESSI will provide the first hard X-ray imaging spectroscopy, the first high-resolution spectroscopy of solar gamma-ray lines from a spacecraft, the first imaging above 100 keV, and the first imaging of solar gamma- ray lines. The gamma-ray imaging spectroscopy will provide the first information on the spatial distribution of energetic (>1 MeV) protons, heavy ions, and relativistic electrons, and the first information on the angular distribution of the energetic ions. It will also provide detailed information on elemental abundances for both the accelerated ions and the ambient ions in the interaction region. HESSI uses Fourier-transform imaging spectroscopy to cover the broad energy range from soft X-rays (2 keV) to gamma-rays (20 MeV) with spatial resolutions down to 2 arcseconds and spectral resolutions down to 1 keV. This capability is achieved with 12 bi-grid rotating modulation collimators located in front of a corresponding set of 12 pairs of cooled germanium and silicon (Si(Li)) detectors to provide the wide spectral coverage. HESSI has been selected by NASA as an alternate Medium-class Explorer (MIDEX) mission, for launch in the year 2000. If it does not get funded as a flight mission, it will be descoped and proposed at a Small Explorer mission for launch in 2000 at half the MIDEX cost.