We consider three related classifications of cellularautomata: the first is based on the complexity of languagesgenerated by clopen partitions of the state space, i.e. on thecomplexity of the factor subshifts; the second is based onthe concept of equicontinuity and it is a modification of theclassification introduced by Gilman [9]. The third oneis based on the concept of attractors and it refines theclassification introduced by Hurley [16]. We showrelations between these classifications and give examples ofcellular automata in the intersection classes. In particular, weshow that every positively expansive cellular automaton isconjugate to a one-sided subshift of finite type and that everytopologically transitive cellular automaton is sensitive toinitial conditions. We also construct a cellular automaton withminimal quasi-attractor, whose basin has measure zero, answeringa question raised in Hurley [16].