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Artificial neural networks (ANNs) are essential tools in machine learning that have drawn increasing attention in neuroscience. Besides offering powerful techniques for data analysis, ANNs provide a new approach for neuroscientists to build models for complex behaviors, heterogeneous neural activity and circuit connectivity, as well as to explore optimization in neural systems, in ways that traditional models are not designed for. In this pedagogical Primer, we introduce ANNs and demonstrate how they have been fruitfully deployed to study neuroscientific questions. We first discuss basic concepts and methods of ANNs. Then, with a focus on bringing this mathematical framework closer to neurobiology, we detail how to customize the analysis, structure, and learning of ANNs to better address a wide range of challenges in brain research. To help the readers garner hands-on experience, this Primer is accompanied with tutorial-style code in PyTorch and Jupyter Notebook, covering major topics.