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The theoretical part of this work deals with evolutionary algorithms (EA), neural networks (NN) and their synthesis in the form of neuroevolution. From a practical point of view, the aim of the work is to show the application of neuroevolution on two different tasks. The first task is the evolutionary design of the convolutional neural network (CNN) architecture that would be able to classify handwritten digits (from the MNIST dataset) with a high accurancy. The second task is the evolutionary optimization of neurocontroller for a simulated Falcon 9 rocket landing. Both tasks are computationally demanding and therefore have been solved on a supercomputer. As a part of the first task, it was possible to design such architectures which, when properly trained, achieve an accuracy of 99.49%. It turned out that it is possible to automate the design of high-quality architectures with the use of neuroevolution. Within the second task, the neuro-controller weights have been optimized so that, for defined initial conditions, the model of the Falcon booster can successfully land. Neuroevolution succeeded in both tasks.