Projections of multivariate climate extremes require methodological approaches that can maintain relationships among variables. Here we apply a piecewise multivariate quantile mapping approach to temperature and humidity projections from a subset of models from the Fifth Coupled Model Intercomparison Project and analyze the resulting climatology of extreme heat days (EHDs) with explicit consideration of the prevailing humidity. The piecewise multivariate bias correction method shows good fidelity in reproducing the frequency of different types of extreme heat events in the historical period. Projections for U.S. regions and individual cities for both the middle and end of the century are characterized by increases in the frequency of EHDs, and especially those characterized by high humidity. For many regions and individual cities, there is no overlap between the frequency of high‐humidity EHDs in general circulation model ensembles from the historical and future periods, indicating that increases in extreme heat are robust. Analysis of 500‐mb height, sea‐level pressure, and low‐level circulation composites for historical and future periods indicates that the Fifth Coupled Model Intercomparison Project models reproduce basic large‐scale circulation features associated with EHDs in U.S. regions and that future changes in extreme heat are related primarily to large‐scale warming rather than enhancement of regional circulation anomalies.