We present experimental and theoretical results concerning the spectral moments of Rayleigh-Brillouin light scattering spectra of pure liquids and gas mixtures. First we derive a general equation for the second and fourth moment of the dynamic structure factor. Second, we derive explicit expressions for these moments for the Gaussian limit and models that predict Lorentzian line shapes. Our experimental results for the normalized second moment agree with the theoretical expressions. The n = 3 sum rule seems to be satisfied. The results for the normalized fourth moment do not agree with theory, but this can be shown to be caused by frequency cutoff effects in the spectrum. The contribution of the instrumental function to the spectral moments is also investigated. Our results indicate that in the analysis of Rayleigh-Brillouin light scattering spectra use can be made of higher order sum rules (at least up to n = 3), provided that the free spectral range of the interferometer is chosen large enough. This would allow for a significant reduction in the number of fitting parameters needed to analyze such a light scattering spectrum.