  We study integrable spin chains and quantum and classical cellular automata
with interaction range $\ell\ge 3$. This is a family of integrable models for
which there was no general theory so far. We develop an algebraic framework for
such models, generalizing known methods from nearest neighbor interacting
chains. This leads to a new integrability condition for medium range
Hamiltonians, which can be used to classify such models. A partial
classification is performed in specific cases, including $U(1)$-symmetric three
site interacting models, and Hamiltonians that are relevant for
interaction-round-a-face models. We find a number of models which appear to be
new. As an application we consider quantum brickwork circuits of various types,
including those that can accommodate the classical elementary cellular automata
on light cone lattices. In this family we find that the so-called Rule150 and
Rule105 models are Yang-Baxter integrable with three site interactions. We
present integrable quantum deformations of these models, and derive a set of
local conserved charges for them. For the famous Rule54 model we find that it
does not belong to the family of integrable three site models, but we can not
exclude Yang-Baxter integrability with longer interaction ranges.
