The energy resolution per bandwidth E R is a figure of merit that combines the field resolution, bandwidth or duration of the measurement, and size of the sensed region. Several very different dc magnetometer technologies approach E R = , while to date none has surpassed this level. This suggests a technology-spanning quantum limit, a suggestion that is strengthened by model-based calculations for nitrogen-vacancy centres in diamond, for dc SQUID sensors, and for optically-pumped alkali-vapor magnetometers, all of which predict a quantum limit close to E R = .Here we review what is known about energy resolution limits, with the aim to understand when and how E R is limited by quantum effects. We include a survey of reported sensitivity versus size of the sensed region for a dozen magnetometer technologies, review the known model-based quantum limits, and critically assess possible sources for a technology-spanning limit, including zero-point fluctuations, magnetic self-interaction, and quantum speed limits. Finally, we describe sensing approaches that appear to be unconstrained by any of the known limits, and thus are candidates to surpass E R = .arXiv:1905.00618v1 [quant-ph]