We investigate the mean dimension of a cellular automaton (CA for short) with a compact non-discrete space of states. A formula for the mean dimension is established for (near) strongly permutative, permutative algebraic and unit one-dimensional automata. In higher dimensions, a CA permutative algebraic or having a spaceship has infinite mean dimension. However, building on Meyerovitch's example [Mey08], we give an example of algebraic surjective cellular automaton with positive finite mean dimension. Contents 1. Introduction 1 2. Background on mean dimension 2 3. Natural extension (X T , T) and skew-product 4 4. General one-dimensional cellular automata 8 5. Permutative one-dimensional CA 10 6. Unit CA 12 7. Algebraic CA 15 8. Higher-dimensional CA having a spaceship 18 Appendix A. Zero metric mean dimension 19 Appendix B. The natural extension of general cellular automata 20 References 21