Nowadays, artificial neural networks are finding more and more applications in various fields of human activity. This paper reviews the current state of research and development in the field of application of artificial neural networks for solving problems in acoustic non-destructive testing. The areas of application of neural networks include processing of inspection data, flaw detection, determination of flaw parameters, and determination of material properties. Test data processing includes increasing the signal-to-noise ratio of results, data compression, restoration and processing of flaw images. Within flaw parameter determination, flaw sizing and flaw type identification tasks have been solved using artificial neural networks. One of the most challenging problems related to the introduction of artificial neural networks in acoustic nondestructive testing is the small amount of inspection data that complicates the training of artificial networks. The ways to solve this problem are data exchange between organizations and standardization of inspection results, data augmentation and application of computer modeling.