There is long-standing scientific interest in understanding purposeful movement by animals and humans. Traditionally, collecting data on individual moving entities was difficult and time-consuming, limiting scientific progress. The growth of location-aware and other geospatial technologies for capturing, managing and analyzing moving objects data are shattering these limitations, leading to revolutions in animal movement ecology and human mobility science. Despite parallel transitions towards massive individual-level data collected automatically via sensors, there is little scientific cross-fertilization across the animal and human divide. There are potential synergies from converging these separate domains towards an integrated science of movement. This paper discusses the data-driven revolutions in the animal movement ecology and human mobility science, their contrasting worldviews and, as examples of complementarity, transdisciplinary questions that span both fields. We also identify research challenges that should be met to develop an integrated science of movement trajectories.