Bar-Natan has introduced a diagrammatic 'picture world' construction
of tangle homology.  A complex of surfaces whose differential is given
by cobordisms with corners is associated to each tangle diagram.  To
get a computable algebraic complex one must apply an appropriate TQFT
with corners to this picture world.  We will explain how open-closed
TQFTs (introduced by Moore and Segal) provide one approach to
algebraically representing the picture world.  Nontrivial examples
extending Khovanov's original homology, as well as the modifications
of Bar-Natan and Lee, will be supplied. Because of the Cardy
condition, an axiom in the definition of an open-closed TQFT, these
extensions only work in finite characteristic. These tangle homologies
differ from the algebraic tangle homology theory introduced by
Khovanov in that these invariants are monoidal -- they map the
disjoint union of tangles to the tensor product of their respective
complexes (Joint work with Hendryk Pfeiffer)