The development of femtosecond pulse lasers has made it possible to generate and detect terahertz (THz) waves with frequencies between 0.1 THz and 10 THz. Terahertz ultrafast spectroscopy based on terahertz pulses is widely used in the study of physical properties for various materials. Due to the characteristic energy of superconductivity being mostly located in the terahertz frequency range, terahertz ultrafast spectroscopy has unique advantages in probing the excitation of various particles and quasi-particles in superconductors. This paper begins with an introduction to terahertz ultrafast spectroscopy and several related data analysis methods. It then reviews the application of terahertz ultrafast spectroscopy in high-temperature superconductors in detail, including the optical properties of equilibrium states, quasi-particle excitation, superconducting modulation, Higgs response, photo-induced superconductivity, and terahertz radiation from high-temperature superconductors. Terahertz ultrafast spectroscopy provides new insights into the rich physical phenomena in high-temperature superconductors and opens up new avenues for the development of terahertz optoelectronic devices based on high-temperature superconductors.