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How do cellular automata behave in the limit of a very large number of cells? Is there a continuum limit with simple properties? We attack this problem by mapping certain classes of automata to quantum field theories for which powerful methods exist for this type of problem. Indeed, many cellular automata admit an interpretation in terms of fermionic particles. Reversible automata on space-lattices with a local updating rule can be described by a partition function or Grassmann functional integral for interacting fermions moving in this space. We discuss large classes of automata that are equivalent to discretized fermionic quantum field theories with various types of interactions. Two-dimensional models include relativistic Thirring or Gross-Neveu type models with abelian or non-abelian continuous global symmetries, models with local gauge symmetries, and spinor gravity with local Lorentz symmetry as well as diffeomorphism invariance in the (naive) continuum limit. The limit of a very large number of cells needs a probabilistic description. Probabilistic cellular automata are characterized by a probability distribution over initial bit-configurations. They can be described by the quantum formalism with wave functions, density matrix and non-commuting operators associated to observables, which are the same for the automata and associated fermionic quantum theories. This formalism is crucial for a discussion of concepts as vacuum states, spontaneous symmetry breaking, coarse graining and the continuum limit for probabilistic cellular automata. In particular, we perform explicitly the continuum limit for an automaton that describes a quantum particle in a potential for one space dimension.