Based on the non-Kolmogorov spectrum, using the extended Huygens-Fresnel principle, the analytical expressions for the spectrumof Gaussian-Schell model (GSM) beams propagating through non-Kolmogorov atmospheric turbulence are derived and used to study the influence of non-Kolmogorov atmospheric turbulence on spectral changes of GSM beams. It is shown that there exist spectral shifts (blueshift and redshift) and spectral transition of GSM beams propagating through non-Kolmogorov atmospheric turbulence. The spectral transition depends on the off-axis distance r, the general exponenta, the general structure constant [C2n], the inner scale l0, the outer scale L0, and the propagation distance z. With the increment of the general exponenta and the inner scale l0, as well as the decrement of the general structure constant [C2n], the spectral transition magnitude [Δ] will decrease and the critical position of spectral transition zc will increase. This work will provide theoretical model and computational basis for researches such as free space optical communications.