Pottery is one of the most commonly recovered artefacts from archaeological sites. Despite more than a century of relative dating based on typology and seriation1, accurate dating of pottery using the radiocarbon dating method has proven extremely challenging owing to the limited survival of organic temper and unreliability of visible residues2-4. Here we report a method to directly date archaeological pottery based on accelerator mass spectrometry analysis of 14C in absorbed food residues using palmitic (C16:0) and stearic (C18:0) fatty acids purified by preparative gas chromatography5-8. We present accurate compound-specific radiocarbon determinations of lipids extracted from pottery vessels, which were rigorously evaluated by comparison with dendrochronological dates9,10 and inclusion in site and regional chronologies that contained previously determined radiocarbon dates on other materials11-15. Notably, the compound-specific dates from each of the C16:0 and C18:0 fatty acids in pottery vessels provide an internal quality control of the results6 and are entirely compatible with dates for other commonly dated materials. Accurate radiocarbon dating of pottery vessels can reveal: (1) the period of use of pottery; (2) the antiquity of organic residues, including when specific foodstuffs were exploited; (3) the chronology of sites in the absence of traditionally datable materials; and (4) direct verification of pottery typochronologies. Here we used the method to date the exploitation of dairy and carcass products in Neolithic vessels from Britain, Anatolia, central and western Europe, and Saharan Africa.