  Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures,
due to work of Hassett, and via Kuznetsov's K3 category A. The relation between
these two viewpoints has recently been elucidated by Addington and Thomas.
  In this paper, both aspects are studied further and extended to twisted K3
surfaces, which in particular allows us to determine the group of
autoequivalences of A for the general cubic fourfold. Furthermore, we prove
finiteness results for cubics with equivalent K3 categories and study periods
of cubics in terms of generalized K3 surfaces.
