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 relevant to optical communications, imaging, sensing, and the design of new materials and lasers. In the following section, we put a special emphasis on how to accurately evaluate the complexity of neural networks in the context of the transition from algorithms to hardware implementation. The introduced complexity characteristics are used to analyze the applications of neural networks in optical communications, as a specific, albeit highly important example, comparing those with some benchmark signal processing methods. We combine the description of the well-known model compression strategies used in machine learning, with some novel techniques introduced recently in optical applications of neural networks. It is important to stress that although our focus in this tutorial-review is on photonics, we believe that the methods and techniques presented here can be handy in a much wider range of scientific and engineering applications.