  The paper contains an essentially self-contained treatment of Khovanov
homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual
knots and virtual knot cobordisms which directly applies to classical knot and
classical knot cobordisms. To do so, we give an alternate formulation for the
Manturov definition of Khovanov homology for virtual knots and links with
arbitrary coefficients. This approach uses cut loci on the knot diagram to
induce a conjugation operator in the Frobenius algebra. We then discuss the
implications of the maps induced in the aforementioned theory to the universal
Frobenius algebra for virtual knots. Next we show how one can apply the Karoubi
envelope approach of Bar-Natan and Morrison on abstract link diagrams with
cross cuts to construct the canonical generators of the Khovanov-Lee homology.
Using these canonical generators we derive a generalization of the Rasmussen
invariant for virtual knot cobordisms and furthermore generalize Rasmussen's
result on the slice genus for positive knots to the case of positive virtual
knots. It should also be noted that this generalization of the Rasmussen
invariant provides an easy to compute obstruction to knot cobordisms in $S_g
\times I \times I$ in the sense of Turaev.
