The higher-order intensity moments of optical beams propagating through atmospheric turbulence are studied in the paper. The method to derive higher-order intensity moments in atmospheric turbulence is proposed, and the simple expressions for intensity moments up to the fourth-order are derived. The results obtained in this paper are general, which can reduce to higher-order intensity moments of an arbitrary optical beam propagating in both free space and turbulence. Taking the Gaussian beam for example, the propagation of the kurtosis parameter in atmospheric turbulence is studied. It is shown that the kurtosis parameter of Gaussian beams is not a propagation invariant in atmospheric turbulence, which depends on propagation distance, waist width, inner and outer scales of turbulence and refraction index structure constant. This result is different from that obtained by using the quadratic approximation of Rytov’s phase structure function or the strong fluctuation condition of turbulence. The reasonable explanations for the differences are given in this paper.