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Annual balances of eight alpine glaciers were slightly negative for 1961–90 and highly negative for 1991–2018. We explain this by changes in positive degree-day sums and summer temperatures extrapolated to the median altitudes of the glaciers. We test a new way of calculating degree-day sums that performs better than the traditional method which used daily mean temperatures. Annual degree-day sums are highly correlated with May–September temperatures as suggested in 1866 by Karl von Sonklar. We find moderate correlations between annual balances and degree-day sums, and with May–September temperatures. Calculated degree-day factors for the eight glaciers cover the reported range for snow and ice ablation, while the temperature sensitivity of annual balance is from −0.4 to −1.0 m w.e. for a +1°C temperature change. We accurately predict mean balances for 1991–2018 using May–September temperatures in regression models calibrated for 1961–90. May–September temperatures in the Alps have already increased ~+3°C since 1880 and, if temperatures continue to rise, these glaciers will shrink rapidly. As annual balances are already negative for present-day temperatures, these glaciers will not be 'safe' under the further temperature increase permitted by the Paris Agreement. [ABSTRACT FROM AUTHOR]