Social networks are becoming larger and more complex as new ways of collecting social interaction data arise (namely from online social networks, mobile devices sensors, ...). These networks are often large-scale and of high dimensionality. Therefore, dealing with such networks became a challenging task. An intuitive way to deal with this complexity is to resort to tensors. In this context, the application of tensor decomposition has proven its usefulness in modelling and mining these networks: it has not only been applied for exploratory analysis (thus allowing the discovery of interaction patterns), but also for more demanding and elaborated tasks such as community detection and link prediction. In this work, we provide an overview of the methods based on tensor decomposition for the purpose of analysing time-evolving social networks from various perspectives: from community detection, link prediction and anomaly/event detection to network summarization and visualization. In more detail, we discuss the ideas exploited to carry out each social network analysis task as well as its limitations in order to give a complete coverage of the topic. © 2020, Springer Nature B.V.