Wet bulb globe temperature (WBGT) is a widely applied heat stress index. However, most applications of WBGT within the heat stress impacts literature do not use WBGT at all, but one of the ad hoc approximations, typically the simplified WBGT (sWBGT) or the environmental stress index (ESI). Surprisingly little is known about how well these approximations work for the global climate and climate change settings that they are being applied to. Here we assess the bias distribution as a function of temperature, humidity, wind speed and radiative conditions of both sWBGT and ESI relative to a well-validated, explicit physical model for WBGT developed by Liljegren, within an idealized context and the more realistic setting of ERA5 reanalysis data. sWBGT greatly overestimates heat stress in hot-humid areas. ESI has much smaller biases in the range of standard climatological conditions. However, both metrics may substantially underestimate extreme heat especially over subtropical dry regions. These systematic biases demonstrate that sWBGT-derived estimates of heat stress and its health and labor consequences are significantly overestimated over much of the world today. We recommend discontinuing the use of sWBGT. ESI may be acceptable for assessing average heat stress or integrated impact over a long period like a year, but less suitable for health applications, extreme heat stress analysis, or as an operational index for heat warning, heatwave forecasting or guiding activity modification at workplace. Nevertheless, Liljegren’s approach should be preferred over these ad hoc approximations and we provide a Python implementation to encourage its widespread use.