The temporal structure of heat waves having substantial human impact varies widely, with many featuring a compound structure of hot days interspersed with cooler breaks. In contrast, many heat wave definitions employed by meteorologists include a continuous threshold-exceedance duration criterion. This study examines the hazard of these diverse sequences of extreme heat in the present, and their change with global warming. We define compound heat waves to include those periods with additional hot days following short breaks in heat wave duration. We apply these definitions to analyze daily temperature data from observations, NOAA Geophysical Fluid Dynamics Laboratory global climate model simulations of the past and projected climate, and synthetically generated time series. We demonstrate that compound heat waves will constitute a greater proportion of heat wave hazard as the climate warms and suggest an explanation for this phenomenon. This result implies that in order to limit heat-related mortality and morbidity with global warming, there is a need to consider added vulnerability caused by the compounding of heat waves.