Abstract: Turbulent flows are ubiquitous in aerospace, oceanography, meteorology, industrial processes and many other fields. In recent years, with the rapid development of modern machine-learning (ML) methods, various neural network (NN)-based turbulence modeling approaches have been proposed and validated. Among these approaches, directly modeling the unsteady temporal evolution of turbulent flow fields through NNs to replace the computationally expensive computational fluid dynamics (CFD) simulations has become a hot research direction recently, and the neural operator (NO) method is becoming an important tool in this field. The current paper focuses on ML-driven flow field prediction methods and, based on the recent works of our research group, provides an overview of the recent advancements in neural operator methods for rapid predictions of turbulent flows. First, we briefly summarize the application directions of ML techniques in fluid mechanics, with a focus on introducing several commonly used ML frameworks for the direct prediction of flow field evolution using neural networks. Next, integrating our group’s research efforts in recent years, we highlight the progress of neural operator methods in the field of turbulence prediction. Specifically, we discuss the research advancements in the application of Fourier neural operators (FNO), Transformer neural operators and physics informed neural operators (PINO) to the flow field prediction in homogeneous isotropic turbulence, turbulent mixing layer, Rayleigh-Taylor turbulence, and turbulent channel flows. In the numerical experiments for various types of turbulence, we found that well-trained neural operator models exhibit higher computational accuracy and efficiency compared to traditional large-eddy simulation (LES)-based CFD methods. These results demonstrate the enormous potential of neural operator-based ML methods in the field of turbulence prediction. Finally, the paper discusses future research directions worthy of attention, including the fast prediction of more complex turbulent flows, the model generalization capability, the embedding of fluid physics, and the theoretical analysis of neural operators.