Heat waves are among the most dangerous climate-related hazards, and they are projected to increase in frequency and intensity over the coming century. Exposure to heat waves is a function of the spatial distribution of physical events and the corresponding population distribution, and future exposure will be impacted by changes in both distributions. In this work we project future exposure using ensembles of climate projections that account for the urban heat island effect, for two alternative emission scenarios (RCP4.5/RCP8.5) and two alternative population and urbanization (SSP3/SSP5) outcomes. We characterize exposure at the global, regional, and grid-cell level; estimate the exposure that would be avoided by mitigating future levels of climate change (to RCP4.5); and quantify the dependence of exposure on population outcomes. We find that climate change is a stronger determinant of exposure than demographic change in these scenarios, with a global reduction in exposure of over 50% under a lower emissions pathway, while a slower population growth pathway leads to roughly 30% less exposure. Exposure reduction varies at the regional level, but in almost all cases the RCP remains more influential than the SSP. Uncertainty in outcomes is dominated by inter-annual variability in heat extremes (relative to variability across initial condition ensemble members). For some regions this variability is large enough that a reduction in annual exposure is not guaranteed in each individual year by following the lower forcing pathway. Finally, we find that explicitly considering the urban heat island effect and separate urban and rural heat extremes and populations can substantially influence results, generally increasing projected exposure.