Laser-based primary thermometry was initiated almost 15 years ago by the proposal to determine the absolute temperature of a gas at thermodynamic equilibrium through the Doppler width of an associated absorption transition, exploiting the potentially very accurate measurement of an optical frequency to infer the elusive thermal energy of a molecular or atomic absorber. This approach, commonly referred to as Doppler broadening thermometry, has benefited across the years from substantial improvements, of both technical and fundamental nature, eventually reaching an accuracy of about 10 ppm on the temperature determination in the best cases. This is sufficient for Doppler broadening thermometry to play a significant role in the practical realization of the new kelvin, which follows the 2019's redefinition from a fixed value of the Boltzmann constant, and to tackle the challenge, among others, to quantify and possibly fix systematic uncertainties of the international ITS-90 temperature scale. This paper reviews and comparatively analyses methods and results achieved so far in the field of laser-based primary thermometry, also including spectroscopic approaches that leverage the temperature-dependent distribution of line intensities and related absorbances across the ro-vibrational band of a molecular sample. Although at an early stage of development, these approaches show a promising degree of robustness with respect to the choice of the line-shape model adopted for the fitting of the absorption spectra, which is a delicate aspect for all laser-based thermometers. We conclude identifying possible technical and scientific evolution axes of the current scenario.