Drifters are freely-drifting satellite-tracked instruments deployed in our oceans to better understand oceanic currents and circulation. In this talk we present a stochastic spatiotemporal model which describes the trajectories that drifters follow. The modelling challenge is that data moves in both time and space, referred to as a "Lagrangian" perspective. Our spatiotemporal model captures effects that are nonstationary, anisotropic, periodic, and non-Markovian or fractal in memory. We use our model to analyse the entire Global Drifter Program database of observations since 1979, constituting over 70 million data points from over 20,000 drifters. I will also briefly discuss how parameter estimation is performed in a computationally-efficient manner, where we propose a novel bias-corrected form of the Whittle likelihood. This is joint work with Sofia Olhede and Arthur Guillaumin (UCL), Jonathan Lilly and Jeffrey Early (NWRA, Seattle), Rick Lumpkin and Renellys Perez (NOAA), and Shane Elipot (Miami).