  Emergence is a profound subject that straddles many scientific disciplines,
including the formation of galaxies and how consciousness arises from the
collective activity of neurons. Despite the broad interest that exists on this
concept, the study of emergence has suffered from a lack of formalisms that
could be used to guide discussions and advance theories. Here we summarise,
elaborate on, and extend a recent formal theory of causal emergence based on
information decomposition, which is quantifiable and amenable to empirical
testing. This theory relates emergence with information about a system's
temporal evolution that cannot be obtained from the parts of the system
separately. This article provides an accessible but rigorous introduction to
the framework, discussing the merits of the approach in various scenarios of
interest. We also discuss several interpretation issues and potential
misunderstandings, while highlighting the distinctive benefits of this
formalism.
