Compared with coherent laser beams, partially coherent beams have advantages of effectively reducing turbulence-induced extra beam spreading, beam wander and intensity scintillation on propagation through turbulent atmosphere, and have promising applications in free-space optical communications, laser radar and remote sensing. Recently, more and more attention was paid to the propagation of partially coherent beams through turbulent atmosphere. In this article, we first review historically the research progress of the propagation of partially coherent beams in atmospheric turbulence. And we describe in detail the basic theory for the calculation of average intensity, second-order moment and scintillation index of partially coherent beams in turbulence based on the extended Huygens-Fresnel principle and Rytov method. We also present a phase screen method of numerically simulating the propagation of coherent beams through turbulent atmosphere, and then extend such a method to treating the propagation of partially coherent beams.