This paper is introductory to the Theme Issue of Philosophical Transactions, which is devoted to the lagrangian description of fluid motions. In much of the literature of the last few decades attention has been focused especially on the eulerian description, in which the time-dependent flow is observed at a given position. This is an important aspect of many experiments, in which velocity and pressure fields are measured at some point. On the other hand, many flow visualizations direct the attention of the observer to the motions of individual particles, and to the phenomena associated with such observations. The lagrangian picture is appropriate for such experiments; new and different phenomena can be seen and understood in a rather simpler manner than in the eulerian description. The basic ideas of the lagrangian picture, and some applications, are therefore laid out in this paper.