Recent years have been marked with the fast-pace diversification and increasing ubiquity of machine learning applications. Yet, a firm theoretical understanding of the surprising efficiency of neural networks to learn from high-dimensional data still proves largely elusive. In this endeavour, analyses inspired by statistical physics have proven instrumental, enabling the tight asymptotic characterization of the learning of neural networks in high dimensions, for a broad class of solvable models. This manuscript reviews the tools and ideas underlying recent progress in this line of work. We introduce a generic model -- the sequence multi-index model -- which encompasses numerous previously studied models as special instances. This unified framework covers a broad class of machine learning architectures with a finite number of hidden units, including multi-layer perceptrons, autoencoders, attention mechanisms; and tasks, including (un)supervised learning, denoising, contrastive learning, in the limit of large data dimension, and comparably large number of samples. We explicate in full detail the analysis of the learning of sequence multi-index models, using statistical physics techniques such as the replica method and approximate message-passing algorithms. This manuscript thus provides a unified presentation of analyses reported in several previous works, and a detailed overview of central techniques in the field of statistical physics of machine learning. This review should be a useful primer for machine learning theoreticians curious of statistical physics approaches; it should also be of value to statistical physicists interested in the transfer of such ideas to the study of neural networks.