A molecular hydrogen ion HD⁺, composed of a proton, a deuteron, and an electron, has a rich set of rovibrational transitions that can be theoretically calculated and experimentally measured precisely. Currently, the relative accuracy of the rovibrational transition frequencies of the HD⁺ molecular ions has reached 10^–12. By comparing experimental measurements with theoretical calculations of the HD⁺ rovibrational spectrum, the precise determination of the proton-electron mass ratio, the testing of quantum electrodynamics(QED) theory, and the exploration of new physics beyond the standard model can be achieved. The experiment on HD⁺ rovibrational spectrum has achieved the highest accuracy (20 ppt, 1 ppt = 10^–12) in measuring proton-electron mass ratio. This ppaper comprehensively introduces the research status of HD⁺ rovibrational spectroscopy, and details the experimental method of the high-precision rovibrational spectroscopic measurement based on the sympathetic cooling of HD⁺ ions by laser-cooled Be⁺ ions. In Section 2, the technologies of generating and trapping both Be⁺ ions and HD⁺ ions are introduced. Three methods of generating ions, including electron impact, laser ablation and photoionization, are also compared. In Section 3, we show the successful control of the kinetic energy of HD⁺ molecular ions through the sympathetic cooling, and the importance of laser frequency stabilization for sympathetic cooling of HD⁺ molecular ions. In Section 4, two methods of preparing internal states of HD⁺ molecular ions, optical pumping and resonance enhanced threshold photoionization, are introduced. Both methods show the significant increase of population in the ground rovibrational state. In Section 5, we introduce two methods of determining the change in the number of HD⁺ molecular ions, i.e. secular excitation and molecular dynamic simulation. Both methods combined with resonance enhanced multiphoton dissociation can detect the rovibrational transitions of HD⁺ molecular ions. In Section 6, the experimental setup and process for the rovibrational spectrum of HD⁺ molecular ions are given and the up-to-date results are shown. Finally, this paper summarizes the techniques used in HD⁺ rovibrational spectroscopic measurements, and presents the prospects of potential spectroscopic technologies for further improving frequency measurement precision and developing the spectroscopic methods of different isotopic hydrogen molecular ions.