Thanks to the abrupt advances in semiconductor technologies, particularly in terms of the operating frequency, the last decade has seen various efforts and trials in attempts to achieve high-throughput wireless communications at terahertz (THz) frequencies. Through the use of several potential device technologies, not only III-V heterojunction bipolar and high-electron-mobility transistors but also silicon complementary metal-oxide-semiconductors and photonic technologies, high data rates of 100 Gbps or more have been successfully demonstrated. Meanwhile, the next generation of wireless communications systems, so called 6G, is already being discussed in research communities worldwide. A terabit per second peak data rate is envisioned, which essentially requires huge spectral resources. The recent considerable attention to 6G communications is implicitly focused on THz communications but is more than a technical curiosity unlike in the past. In this article, we briefly review the progress of THz communications made in the last decade and discuss some of the most challenging issues that need to be overcome for future wireless systems at THz frequencies, including not only the 6G network but also short-range connectivity and fixed wireless links.