  This thesis splits into two major parts. The connection between the two parts
is the notion of "categorification" which we shortly explain/recall in the
introduction.
  In the first part of this thesis 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 2, extensions with no
classical analogon, e.g. Bar-Natan's Z/2Z-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.
  In the second part of this thesis (which is based on joint work with Mackaay
and Pan) we use Kuperberg's sl3 webs and Khovanov's sl3 foams to define a new
algebra K_S, which we call the sl3 web algebra. It is the sl3 analogue of
Khovanov's arc algebra H_n.
  We prove that K_S is a graded symmetric Frobenius algebra. Furthermore, we
categorify an instance of q-skew Howe duality, which allows us to prove that
K_S is Morita equivalent to a certain cyclotomic KLR-algebra. This allows us to
determine the split Grothendieck group K_0(K_S), to show that its center is
isomorphic to the cohomology ring of a certain Spaltenstein variety, and to
prove that K_S is a graded cellular algebra.
