Abstract: We report on the latest achievements in the development of superconducting hot-electron bolometer (HEB) mixers for terahertz superheterodyne receivers. We consider application ranges of such receivers and requirements for the basic characteristics of the mixers. Main features of the mixers, such as noise temperature, gain bandwidth, noise bandwidth, and required local-oscillator power, have been improved significantly over the past few years due to intense research work, both in terms of the element fabrication quality and in terms of understanding of the physics of the processes occurring in the HEB mixers. Contacts between the superconducting bridge and the planar antenna play a key role in the mixer operation. Improvement of the quality of the contacts leads simultaneously to a decrease in the noise temperature and an increase in the gain bandwidth of a mixer.