Facilitated by the rapid technological development of the near-space platform stations (NSPS), near-space communication (NS-COM) is envisioned to play a pivotal role in the space-air-ground integrated network (SAGIN) for the sixth generation (6G) communications and beyond. In NS-COM, ultrabroadband wireless connectivities between NSPSs and various airborne/spaceborne platforms are required for a plethora of bandwidth-consuming applications, such as NSPS-based ad hoc networking, the in-flight Internet and the relaying technology. However, such requirement seem to contradict the scarcity of spectrum resources at conventional microwave frequencies, which motivates the exploitation of terahertz (THz) band ranging from 0.1 to 10 THz. Due to huge available bandwidth, THz signals are capable of supporting ultra-high-rate data transmission for NS-COM over 100 Gb/s, which are naturally suitable for the near-space environment with marginal path loss. Against this background, this article provides an extensive investigation on THz-band NS-COM (THz-NS-COM) from a physical-layer perspective. Firstly, we summarize the potential applications of THz communications in the near-space environment, where the corresponding technical barriers are analyzed. Secondly, the channel characteristics of THz-NS-COM and the corresponding modeling strategies are discussed. Thirdly, three essential research directions are investigated to surpass the technical challenges of THz-NS-COM, i.e., robust beamforming for ultramassive antenna array, signal processing algorithms against hybrid distortions, and integrated sensing and communications. Several open problems are also provided to unleash the full potential of THz-NS-COM.