  We introduce the notion of a Khovanov-Floer theory. Roughly, such a theory
assigns a filtered chain complex over Z/2 to a link diagram such that (1) the
E_2 page of the resulting spectral sequence is naturally isomorphic to the
Khovanov homology of the link; (2) this filtered complex behaves nicely under
planar isotopy, disjoint union, and 1-handle addition; and (3) the spectral
sequence collapses at the E_2 page for any diagram of the unlink. We prove that
a Khovanov-Floer theory naturally yields a functor from the link cobordism
category to the category of spectral sequences. In particular, every page
(after E_1) of the spectral sequence accompanying a Khovanov-Floer theory is a
link invariant, and an oriented cobordism in R^3 \times [0,1] between links in
R^3 induces a map between each page of their spectral sequences, invariant up
to smooth isotopy of the cobordism rel boundary.
  We then show that the spectral sequences relating Khovanov homology to
Heegaard Floer homology and singular instanton knot homology are induced by
Khovanov-Floer theories and are therefore functorial in the manner described
above, as has been conjectured for some time. We further show that Szabo's
geometric spectral sequence comes from a Khovanov-Floer theory, and is thus
functorial as well. In addition, we illustrate how our framework can be used to
give another proof that Lee's spectral sequence is functorial and that
Rasmussen's invariant is a knot invariant. Finally, we use this machinery to
define some potentially new knot invariants.
