The study of high-precision spectroscopy for hydrogen molecular ions enables the determination of fundamental constants, such as the proton-to-electron mass ratio, the deuteron-to-electron mass ratio, the Rydberg constant, and the charge radii of proton and deuteron. This can be accomplished through a combination of high precision experimental measurements and theoretical calculations. The spectroscopy of hydrogen molecular ions reveals abundant hyperfine splittings, necessitating not only an understanding of rovibrational transition frequencies but also a thorough grasp of hyperfine structure theory to extract meaningful physical information from the spectra. This article reviews the history of experiments and theories related to the spectroscopy of hydrogen molecular ions, with a particular focus on the theory of hyperfine structure. As far back as the second half of the last century, the hyperfine structure of hydrogen molecular ions was described by a comprehensive theory based on its leading-order term, known as the Breit-Pauli Hamiltonian. Thanks to the advancements in non-relativistic quantum electrodynamics (NRQED) at the beginning of this century, a systematic development of next-to-leading-order theory for hyperfine structure has been achieved and applied to $\text{H}_2^+$ and $\text{HD}^+$ in recent years, including the establishment of the $m\alpha^7\ln(\alpha)$ order correction. For the hyperfine structure of $\text{H}_2^+$, theoretical calculations show good agreement with experimental measurements after decades of work. However, for HD ⁺, discrepancies have been observed between measurements and theoretical predictions that cannot be accounted for by the theoretical uncertainty in the non-logarithmic term of the $m\alpha^7$ order correction. To address this issue, additional experimental measurements are needed for mutual validation, as well as independent tests of the theory, particularly regarding the non-logarithmic term of the $m\alpha^7$ order correction.