A general treatment of nuclear magnetic resonance (NMR) spectra under magic-angle spinning (MAS) conditions is provided that is applicable both to homogeneously and inhomogeneously broadened lines. It is based on a combination of Floquet theory and perturbation theory, and allows the factorization of the spin system response into three factors that describe different aspects of the resulting MAS spectrum. The first factor directly reflects the Floquet theorem and describes the appearance of sidebands, The other two terms give the integral intensities of the resulting sidebands and their line shapes and depend on the specific features of the considered interaction. The analytical form of these two factors is derived for multi-spin dipolar interactions under fast MAS. The leading term in the expansion of the integral intensities involves products of only two spin operators whereas the linewidths, which are found to be different for the different sideband orders, are determined predominantly by three-spin terms. The higher-spin contributions in both cases scale with increasing powers of the inverse rotor frequency and thus becomes less and less important when approaching the limit of fast spinning. From numerical simulations and the analysis of experimental MAS NMR spectra it was found that for typical spin systems, spinning frequencies of the order of the strongest couplings are sufficient to allow the analysis of the sideband intensities within the approximation of two-spin terms. This scaling of the different contributions together with the strong distance dependence of the dipolar interaction thus leads to a considerable simplification in the fast spinning limit and provides the basis for using the dipolar interaction in high-resolution MAS spectra to obtain local structural information.