Many scientific domains exhibit phenomena that seem to be “more than the sum of their parts”; for example, flocks seem to be more than a mere collection of birds, and consciousness seems more than electric impulses between neurons. But what does it mean for a physical system to exhibit emergence? The literature on this topic contains various conflicting approaches, many of which are unable to provide quantitative, falsifiable statements. Having a rigorous, quantitative theory of emergence could allow us to discover the exact conditions that allow a flock to be more than individual birds, and to better understand how the mind emerges from the brain. Here we provide exactly that: a formal theory of what constitutes causal emergence, how to measure it, and what different “types” of emergence exist. To do this, we leverage recent developments in information dynamics—the study of how information flows through and is modified by dynamical systems. As part of this framework, we provide a mathematical definition of causal emergence, and also practical formulae for analysing empirical data. Using these, we are able to confirm emergence in the iconic Conway’s Game of Life, in certain flocking patterns, and in representations of motor movements in the monkey’s brain.