We extend Bar-Natan's cobordism based categorification of the Jones
polynomial to virtual links. Our topological complex allows a direct extension
of the classical Khovanov complex ($h=t=0$), the variant of Lee ($h=0,t=1$) and
other classical link homologies. We show that our construction allows, over
rings of characteristic two, extensions with no classical analogon, e.g.
Bar-Natan's $\mathbb{Z}/2$-link homology can be extended in two non-equivalent
ways.
Our construction is computable in the sense that one can write a computer
program to perform calculations, e.g. we have written a Mathematica based
program.
Moreover, we give a classification of all unoriented TQFTs which can be used
to define virtual link homologies from our topological construction.
Furthermore, we prove that our extension is combinatorial and has semi-local
properties. We use the semi-local properties to prove an application, i.e. we
give a discussion of Lee's degeneration of virtual homology.