Abstract The ‘Snowball Earth’ model of Hoffman et al. [Science 281 (1998) 1342] has stimulated renewed interest in the causes of glaciation in Earth history and the sedimentary, stratigraphic and geochemical response. The model invokes catastrophic global Neoproterozoic refrigerations when oceans froze, ice sheets covered the tropics and global temperatures plummeted to −50 °C. Each event is argued to be recorded by tillites and have lasted up to 10 million years. Planetary biological activity was arrested only to resume in the aftermath of abrupt and brutal volcanically generated ‘greenhouse’ deglaciations when global temperatures reached +50 °C. The ‘Cambrian explosion’ is regarded by some as a consequence of post-Snowball glacioeustatic flooding of continental shelves. We shall show by a systematic review of the model that it is based on many long standing assumptions of the character and origin of the Neoproterozoic glacial record, in particular, ‘tillites’, that are no longer valid. This paper focusses on the sedimentological and stratigraphic evidence for glaciation in the light of current knowledge of glacial depositional systems. By integrating this analysis with recent understanding of the tectonic setting of Neoproterozoic sedimentary basins, an alternative ‘Zipper-rift’ hypothesis for Neoproterozoic glaciations is developed. The ‘Zipper-rift’ model emphasises the strong linkage between the first-order reorganisation of the Earth's surface created by diachronous rifting of the supercontinent Rodinia, the climatic effects of uplifted rift flanks and the resulting sedimentary record deposited in newly formed rift basins. Initial fragmentation of Rodinia commenced after about 750 Ma (when the paleo-Pacific Ocean started to form along the western margin of Laurentia) and culminated sometime after 610 Ma (with the opening of the paleo-Atlantic Ocean on the eastern margin of Laurentia). Breakup is recorded by well-defined ‘tectonostratigraphic’ successions that were deposited in marine rift basins. The base of each succession is characterised by coarse-grained synrift strata consisting of mass flow diamictites and conglomerates (many of the ‘tillites’ of the older literature). These facies are interbedded with large olistostromes and contain clastic carbonate debris derived from landsliding of fault scarps along rifted carbonate platforms. Diamictites and conglomerates are dominantly the product of subaqueous mass flow and mixing of coarse and fine sediment populations (the term mixtite has been used in the past). These facies are not uniquely glacial and are produced regardless of climate and latitude. Synrift deposits commonly pass up into thick slope turbidites recording enhanced subsidence and are capped by uppermost shallow marine strata that record a reduction in subsidence rates and overall basin shallowing. Tectonostratigraphic ‘cycles’ can attain thicknesses of several kilometres, but have been commonly misinterpreted as recording globally synchronous ‘glacioeustatic’, falls and rises in sea level. In fact, eustatic sea-level changes in rift basins are suppressed as a result of a strong tectonic control on relative water depths. The great length of newly formed rifted margins around the long perimeter of Laurentia (<20,000 km) ensured that deposition of tectonostratigraphic successions occurred diachronously as in the manner of a zipper between approximately 740 and 610 Ma. Some successions show a definite glacial influence on sedimentation as a consequence of latitude or strong tectonic uplift, but many do not. Regardless, all deposits record a fully functioning hydrological cycle entirely at odds with a supposed fully permafrozen planet. Mapping of those deposits where a definite glacial imprint is apparent indicates that Neoproterozoic glaciation(s) were likely regional or hemispheric in scope and latitudinally constrained. They were perhaps no more severe than other glaciations recorded in Earth history. The regional distribution of ice centres is argued to have been influenced by tectonic topography created by large mantle plumes and by rift shoulder uplift. Paleomagnetic data indicative of tropical glaciation are, in our view, ambiguous because of uncertainty as to when such paleomagnetic characteristics were acquired. A lower solar luminosity may have played a role in lowering snow line elevations and displacing glaciation into latitudes lower than those of Phanerozoic glaciations. Global tectonic and volcanic activity, especially the rapid burial or organic carbon in new rift basins, may explain extreme shifts in C-isotopic values evident in late Neoproterozoic strata.