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This tutorial–review on applications of artificial neural networks in photonics targets a broad audience, ranging from optical research and engineering communities to computer science and applied mathematics. We focus here on the research areas at the interface between these disciplines, attempting to find the right balance between technical details specific to each domain and overall clarity. First, we briefly recall key properties and peculiarities of some core neural network types, which we believe are the most relevant to photonics, also linking the layer’s theoretical design to some photonics hardware realizations. After that, we elucidate the question of how to fine-tune the selected model’s design to perform the required task with optimized accuracy. Then, in the review part, we discuss recent developments and progress for several selected applications of neural networks in photonics, including multiple aspects …