Virtual knot theory is an interesting generalization of knot theory in which ``virtual" crossings are allowed. See Kauffman's Virtual Knot Theory for an introduction. Greg Kuperberg gave a nice topological interpretation of virtual knots in this paper. One reason to be interested in virtual knots and links is that many knot and link invariants generalize to the virtual setting. For example, my naive understanding is that virtual knots are a more natural domain for Vassiliev invariants than knots are. My question is whether anyone knows of examples that demonstrate the utility of virtual knot theory? For example, are there any interesting theorems outside of virtual knot theory that can be most easily proven using virtual knot theory? The papers I have seen seem to pursue VKT for its own sake, but my sense is that such a natural area must be of much wider use.
Added: I just ran across a paper by Rourke that explains a geometric interpretation of "welded links" which are like virtual links but an additional move is allowed. This is a beautiful little paper which explains how welded knot theory corresponds to a theory of certain embedded tori in $\mathbb R^4$. It's really amazing how the Reidemeister moves, both virtual and classical, correspond to isotopies of what Rourke calls toric links. This is a part of Bar-Natan's program mentioned by Theo Johnson-Freyd and Daniel Moskovich below. Dror calls the toric links "flying rings".