Due to its brightness, directionality, and monochromaticity, coherent laser light has been widely used in military defense, medicine, industrial processing, optical communication systems and other fields, playing a vital role in human social progress and economic development. However, with the development of application of laser light, it is found that the one with high coherence will induce some negative effects. Fortunately, it is found that decreasing the coherence can not only keep the original features, but also reduce many negative effects caused by high coherence. Therefore, spatial coherence has gradually become fourth intrinsic property of light that can be optimized for particular tasks, in addition to amplitude, polarization and phase.Laser beams with decreased spatial coherence, called partially coherent beams, and they often have advantages over their coherent counterparts. In the past few decades, researchers only focused on the conventional partially coherent beams, i.e., Gaussian Schell-model beams, and such beams exhibit single and boring propagation property, which cannot meet the increasing demand for laser featrues. Therefore, how to manipulate the propagation features of laser fields to meet the actual demand is particularly important. From the perspective of manipulation of the coherence structures, a new class of light field with prescribed coherence structures can show novel propagation features. Until recently, only a few papers were devoted to partially coherent beams with non-conventional correlations (i.e. non-Gaussian correlated), such as J₀-correlated Schell-model beams and vortex-carrying partially coherent beams. Investigations of such beams were limited due to the difficulty in proving that a given function is, in fact, a mathematically valid correlation function.But in 2007, a powerful new method for designing correlation functions of scalar partially coherent beams was introduced by Gori and Santarsiero, followed in 2009 by a more general method for vector partially coherent beams, allowing a wide variety of novel partially coherent beams to be investigated. Among the classes that have been studied since then are multi-Gaussian correlated Schell-model beams, Laguerre-Gaussian correlated Schell-model beams, Hermite-Gaussian Schell-model beams, and optical coherence lattices. Such beams display many extraordinary and potentially beneficial properties, such as flat-topped and ring-shaped intensity profiles in the far field, self-splitting properties, and lattice-like intensity patterns that form on propagation. And they have useful applications in many areas, such as free-space optical communication, particle trapping, image transmission and image encryption.In this review, we first outline the fundamental theories on constructing scaler and vector partially coherent beams with prescribed correlation structures, and then present detailed description of the beams models and their propagation properties of several typical examples. Finally, we review the methods of experimental generation of partially coherent beams with prescribed correlation structures. We hope our review will stimulate further efforts in this area of research.