A global Earth system model is used to study the relationship between heat waves and surface ozone levels over land areas around the world that could experience either large decreases or little change in future ozone precursor emissions. The model is driven by emissions of greenhouse gases and ozone precursors from a medium-high emission scenario (Representative Concentration Pathway 6.0–RCP6.0) and is compared to an experiment with anthropogenic ozone precursor emissions fixed at 2005 levels. With ongoing increases in greenhouse gases and corresponding increases in average temperature in both experiments, heat waves are projected to become more intense over most global land areas (greater maximum temperatures during heat waves). However, surface ozone concentrations on future heat wave days decrease proportionately more than on non-heat wave days in areas where ozone precursors are prescribed to decrease in RCP6.0 (e.g. most of North America and Europe), while surface ozone concentrations in heat waves increase in areas where ozone precursors either increase or have little change (e.g. central Asia, the Mideast, northern Africa). In the stabilized ozone precursor experiment, surface ozone concentrations increase on future heat wave days compared to non-heat wave days in most regions except in areas where there is ozone suppression that contributes to decreases in ozone in future heat waves. This is likely associated with effects of changes in isoprene emissions at high temperatures (e.g. west coast and southeastern North America, Eastern Europe).