Abstract: Time series causal discovery --- the process of detecting causation between time series --- has its roots in signal processing research from over a half century ago. It has since received active and widespread interest, evolving into different forms across domains such as statistics, dynamical systems and machine learning. Entering this fragmented and constantly evolving research landscape can be daunting. This tutorial provides a unified overview of time series causal discovery across research communities. We introduce the key principles invoked by modern discovery methods, and we discuss their limitations and assumptions. We also discuss how discovery methods are evaluated, benchmarked and fine-tuned. We further discuss how causal discovery for time series is becoming crucial for downstream tasks in agriculture and Earth science, in healthcare, and in artificial intelligence systems. We believe that this tutorial will serve as a valuable resource in navigating the deluge of recent causality research, as well as an invitation to signal processing audiences to contribute to causal problems. To further help readers, we host a companion repository to link to useful code packages and resources at https://github.com/CHAI-UK/Awesome-Time-Series-Causal-Discovery.