  We use a special kind of 2-dimensional extended Topological Quantum Field
Theories (TQFTs), so-called open-closed TQFTs, in order to extend Khovanov
homology from links to arbitrary tangles, not necessarily even. For every plane
diagram of an oriented tangle, we construct a chain complex whose homology is
invariant under Reidemeister moves. The terms of this chain complex are modules
of a suitable algebra A such that there is one action of A or A^op for every
boundary point of the tangle. We give examples of such algebras A for which our
tangle homology theory reduces to the link homology theories of Khovanov, Lee,
and Bar-Natan if it is evaluated for links. As a consequence of the Cardy
condition, Khovanov's graded theory can only be extended to tangles if the
underlying field has finite characteristic. In all cases in which the algebra A
is strongly separable, i.e. for Bar-Natan's theory in any characteristic and
for Lee's theory in characteristic other than 2, we also provide the required
algebraic operation for the composition of oriented tangles. Just as Khovanov's
theory for links can be recovered from Lee's or Bar-Natan's by a suitable
spectral sequence, we provide a spectral sequence in order to compute our
tangle extension of Khovanov's theory from that of Bar-Natan's or Lee's theory.
Thus, we provide a tangle homology theory that is locally computable and still
strong enough to recover characteristic p Khovanov homology for links.
