Abstract : A subshift attractor is a two-sided subshift which is an attractor of a cellular automaton. We prove that each subshift attractor is chain-mixing, contains a configuration which is both F-periodic and $\sigma$-periodic and the complement of its language is recursively enumerable. We prove that a subshift of finite type is an attractor of a cellular automaton iff it is mixing. We identify a class of chain-mixing sofic subshifts which are not subshift attractors. We construct a cellular automaton whose maximal attractor is a non-sofic mixing subshift, answering a question raised in Maass. We show that a cellular automaton is surjective on its small quasi-attractor which is the non-empty intersection of all subshift attractors of all $F^q\sigma^p$, where $q>0$ and $p\in Z$.