The Earth surface thermal infrared (TIR) radiation shows conspicuously an anisotropic behavior just like the bi-directional reflectance of visible and near infrared spectral domains. The importance of thermal radiation directionality (TRD) is being more and more widely recognized in the applications because of the magnitude of the effects generated. The effects of TRD were originally evidenced through experiments in 1962, showing that two sensors simultaneously measuring temperature of the same scene may get significantly different values when the viewing geometry is different. Such effect limits inter-comparison of measurement datasets and land surface temperature (LST) products acquired at different view angles, while raising the question of measurement reliability when used to characterize land surface processes. These early experiments fostered the development of modeling approaches to quantify TRD with the aim of developing a correction for Earth surface TIR radiation. Initiatives for pushing the analysis of TIR data through modeling have been lasted since 1970s. They were initially aimed at mimicking the observed TIR radiance with consideration of canopy structure, component emissivities and temperatures, and Earth surface energy exchange processes. Presently, observing the Earth surface TRD effect is still a challenging task because the TIR status changes rapidly. Firstly, a brief theoretical background and the basic radiative transfer equation are presented. Then, this paper reviews the historical development and current status of observing TRD in the laboratory, in-situ, from airborne and space-borne platforms. Accordingly, the TRD model development, including radiative transfer models, geometric models, hybrid models, 3D models, and parametric models are reviewed for surfaces of water, ice and sea, snow, barren lands, vegetation and urban landscapes, respectively. Next, we introduce three potential applications, including normalizing the LST products, estimating the hemispheric upward longwave radiation using multi-angular TIR observations and separating surface component temperatures. Finally, we give hints and directions for future research work. The last section summarizes the study and stresses three main conclusions.