We introduce in this survey the major concepts, models, and algorithms proposed so far to infer causal relations from observational time series, a task usually referred to as causal discovery in time series. To do so, after a description of the underlying concepts and modelling assumptions, we present different methods according to the family of approaches they belong to: Granger causality, constraint-based approaches, noise-based approaches, score-based approaches, logic-based approaches, topology-based approaches, and difference-based approaches. We then evaluate several representative methods to illustrate the behaviour of different families of approaches. This illustration is conducted on both artificial and real datasets, with different characteristics. The main conclusions one can draw from this survey is that causal discovery in times series is an active research field in which new methods (in every family of approaches) are regularly proposed, and that no family or method stands out in all situations. Indeed, they all rely on assumptions that may or may not be appropriate for a particular dataset.