  Discretizing spacetime is often a natural step towards modelling physical
systems. For quantum systems, if we also demand a strict bound on the speed of
information propagation, we get quantum cellular automata (QCAs). These
originally arose as an alternative paradigm for quantum computation, though
more recently they have found application in understanding topological phases
of matter and have been proposed as models of periodically driven (Floquet)
quantum systems, where QCA methods were used to classify their phases. QCAs
have also been used as a natural discretization of quantum field theory, and
some interesting examples of QCAs have been introduced that become interacting
quantum field theories in the continuum limit. This review discusses all of
these applications, as well as some other interesting results on the structure
of quantum cellular automata, including the tensor-network unitary approach,
the index theory and higher dimensional classifications of QCAs.
