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The HyGAL SOFIA legacy program surveys six hydride molecules -- ArH+, OH+, H2O+, SH, OH, and CH -- and two atomic constituents -- C+ and O -- within the diffuse interstellar medium (ISM) by means of absorption-line spectroscopy toward 25 bright Galactic background continuum sources. This detailed spectroscopic study is designed to exploit the unique value of specific hydrides as tracers and probes of different phases of the ISM, as demonstrated by recent studies with the Herschel Space Observatory. The observations performed under the HyGAL program will allow us to address several questions related to the lifecycle of molecular material in the ISM and the physical processes that impact its phase transition, such as: (1) What is the distribution function of the H2 fraction in the ISM? (2) How does the ionization rate due to low-energy cosmic-rays vary within the Galaxy? (3) What is the nature of interstellar turbulence, and what mechanisms lead to its dissipation? This overview discusses the observing strategy, synergies with ancillary and archival observations, the data reduction and analysis schemes adopted; and presents the first results obtained toward three of the survey targets, W3(OH), W3IRS5 and NGC7538IRS1. Robust measurements of the column densities of these hydrides -- obtained through widespread observations of absorption lines-- help address the questions raised, and there is a timely synergy between these observations and the development of theoretical models, particularly pertaining to the formation of H2 within the turbulent ISM. The provision of enhanced HyGAL data products will therefore serve as a legacy for future ISM studies.