We consider three related classifications of cellular
automata: the first is based on the complexity of languages
generated by clopen partitions of the state space, i.e. on the
complexity of the factor subshifts; the second is based on
the concept of equicontinuity and it is a modification of the
classification introduced by Gilman [9]. The third one
is based on the concept of attractors and it refines the
classification introduced by Hurley [16]. We show
relations between these classifications and give examples of
cellular automata in the intersection classes. In particular, we
show that every positively expansive cellular automaton is
conjugate to a one-sided subshift of finite type and that every
topologically transitive cellular automaton is sensitive to
initial conditions. We also construct a cellular automaton with
minimal quasi-attractor, whose basin has measure zero, answering
a question raised in Hurley [16].