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This review comes at the time when ALMA successfully obtained the first regular observations of both a prominence and a filament. These observations have a spatial resolution of 1-2 arcsec, far better than previous prominence observations in the mm/sub-mm radio domain. The achieved resolution is compatible with the cotemporal ground-based coronagraphic observations in the hydrogen Hα line that accompany the ALMA Band 3 prominence observations. A core part of this review is the description and analysis of these pioneering ALMA observations of a quiescent prominence, focusing on various physical and geometrical properties of the observed prominence fine structures. We also summarize the basic physical processes behind the formation of mm/sub-mm continua under prominence conditions and describe the plasma diagnostics potential of ALMA solar observations. Finally, we discuss future prospects of ALMA prominence observations and the value of coordinated optical and UV spectra and images.