Recent work has revealed intricate connections between non-invertible symmetries and quantum cellular automata (QCAs) in 1+1 dimensions. On the one hand, non-invertible symmetries themselves can be viewed as QCAs acting on abstract spin chains. On the other hand, when restricted to ordinary spin chains, non-invertible symmetries can sometimes be realized only after mixing with ordinary QCAs. In this talk, I will review these recent developments, following work of Corey Jones and collaborators, as well as Kansei Inamura.