Quantitative knowledge of the surface energy balance is essential for the prediction of weather and climate. However, a multitude of studies from around the world indicate that the turbulent heat fluxes are generally underestimated using eddy-covariance measurements, and hence, the energy balance is not closed. This energy-balance-closure problem, which has been heavily covered in the literature for more than 25 years, is the topic of the present review, in which we provide an overview of the potential reason for the lack of closure. We demonstrate the effects of the diurnal cycle on the energy balance closure, and address questions with regard to the partitioning of the energy balance residual between the sensible and the latent fluxes, and whether the magnitude of the flux underestimation can be predicted based on other variables typically measured at micrometeorological stations. Remaining open questions are discussed and potential avenues for future research on this topic are laid out. Integrated studies, combining multi-tower experiments and scale-crossing, spatially-resolving lidar and airborne measurements with high-resolution large-eddy simulations, are considered to be of critical importance for enhancing our understanding of the underlying transport processes in the atmospheric boundary layer.