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We identify an important set of key areas where an advanced observational Ultraviolet capability would have major impact on studies of cosmology and Galaxy formation in the young Universe. Most of these are associated with the Universe at z < 3–4. We address the issues associated with Dark matter evidence in the local Universe and the impact of the Warm-Hot Intergalactic Medium WHIM on the local Baryon count. The motivations to make ultraviolet (UV) studies of supernovae (SNe) are reviewed and discussed in the light of the results obtained so far by means of IUE and HST observations. It appears that UV studies of SNe can, and do lead to fundamental results not only for our understanding of the SN phenomenon, such as the kinematics and the metallicity of the ejecta, but also for exciting new findings in Cosmology, such as the tantalizing evidence for “dark energy” that seems to pervade the Universe and to dominate its energetics. The need for additional and more detailed UV observations is also considered and discussed. Finally we show the enormous importance of the UV for abundance evolution in the Intergalactic Medium (IGM), and the importance of the He II studies to identify re-ionization epochs, which can only be done in the UV. [ABSTRACT FROM AUTHOR]