Orbital angular momentum (OAM) is an important degree of freedom of light. As having unique phase and intensity distribution, OAM-carrying light has wide applications in high capacity optical communication, metrology, imaging, optical tweezers and quantum information processing. In one hand, frequency conversion with OAM-carrying light in nonlinear process shows new optical physics different from that of Gaussian beam. On the other hand, nonlinear optics offers various tool kits to manipulate different content of light, which can be used to generate specific light field for target applications. In this article, the main progress for frequency conversion of OAM-carrying light in quasi-phase matching nonlinear crystals is reviewed, including the conservation, propagation, evolution and interference of OAM-carrying light in frequency conversion, high efficiency nonlinear frequency conversion for OAM-carrying light, realization of frequencey conversion efficiency independent of the input signal modes, frequency conversion of vector vortex beam, and generation of high dimensional OAM entangled states without post selection based on pump beam engineering. Finally, some discussions and outlooks for future research trends in this field are put forward.