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As early as at the end of the 19th century, shortly after mathematical rules describing fluid flow—such as the Navier–Stokes equations—were developed, the idea of using them for flow simulations emerged. However, it was soon discovered that the computational requirements of problems such as atmospheric phenomena and engineering calculations made hand computation impractical. The dawn of the computer age also marked the beginning of computational fluid mechanics and their subsequent popularization made computational fluid dynamics one of the common tools used in science and engineering. From the beginning, however, the method has faced a trade-off between accuracy and computational requirements. The purpose of this work is to examine how the results of recent advances in machine learning can be applied to further develop the seemingly plateaued method. Examples of applying this method to improve various types of computational flow simulations, both by increasing the accuracy of the results obtained and reducing calculation times, have been reviewed in the paper as well as the effectiveness of the methods presented, the chances of their acceptance by industry, including possible obstacles, and potential directions for their development. One can observe an evolution of solutions from simple determination of closure coefficients through to more advanced attempts to use machine learning as an alternative to the classical methods of solving differential equations on which computational fluid dynamics is based up to turbulence models built solely from neural networks. A continuation of these three trends may lead to at least a partial replacement of Navier–Stokes-based computational fluid dynamics by machine-learning-based solutions.