Facilitated by 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 for sixth-generation (6G) communications and beyond. In NS-COM, ultra-broadband wireless connectivity between NSPSs and various airborne/spaceborne platforms is required for a plethora of bandwidth-consuming applications, such as NSPS-based Ad hoc networking, in-flight Internet and relaying technology. However, such requirement seems to contradict with 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, the 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. To this end, this article provides an extensive investigation on the 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. Then the channel characteristics of THz-NS-COM and the corresponding modeling strategies are discussed, respectively. Afterwards, three essential research directions are investigated to surpass the technical barriers of THz-NS-COM, i.e., robust beamforming for ultra-massive 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.