Altmetrics are web-based quantitative impact or attention indicators for academic articles that have been proposed to supplement citation counts. This article reports the first assessment of the extent to which mature altmetrics from Altmetric.com and Mendeley associate with individual article quality scores. It exploits expert norm-referenced peer review scores from the UK Research Excellence Framework 2021 for 67,030+ journal articles in all fields 2014–2017/2018, split into 34 broadly field-based Units of Assessment (UoAs). Altmetrics correlated more strongly with research quality than previously found, although less strongly than raw and field normalized Scopus citation counts. Surprisingly, field normalizing citation counts can reduce their strength as a quality indicator for articles in a single field. For most UoAs, Mendeley reader counts are the best altmetric (e.g., three Spearman correlations with quality scores above 0.5), tweet counts are also a moderate strength indicator in eight UoAs (Spearman correlations with quality scores above 0.3), ahead of news (eight correlations above 0.3, but generally weaker), blogs (five correlations above 0.3), and Facebook (three correlations above 0.3) citations, at least in the United Kingdom. In general, altmetrics are the strongest indicators of research quality in the health and physical sciences and weakest in the arts and humanities.