Martian global climate models that account for evidence of past glaciations reported in tropical to mid-latitude regions of Mars predict accumulation of water ice in Valles Marineris during past periods of high obliquity. Observational evidence for such glaciations is given here. Topographic basement ridges of tectonic origin are common in Valles Marineris, and display sackung features, an assemblage of tectonic patterns that are diagnostic of deep-seated gravitational slope deformation. This deformation is most easily explained by paraglacial ridge failure subsequent to ridge wall debuttressing and decohesion following the retreat of glaciers. This interpretation is supported by extensive bibliographic analysis of sackung triggers on Earth, by morphological evidence of subglacial erosion of the lower parts of Valles Marineris wallslopes, of periglacial erosion of their upper parts and by the presence of various types of glacial landforms on the floors of Valles Marineris troughs. The age of these equatorial glaciations is found to be older than 1.4Gy and younger than 3.5Gy.