  Virtual knots, defined by Kauffman, provide a natural generalization of
classical knots. Most invariants of knots extend in a natural way to give
invariants of virtual knots. In this paper we study the fundamental groups of
virtual knots and observe several new and unexpected phenomena.
  In the classical setting, if the longitude of a knot is trivial in the knot
group then the group is infinite cyclic. We show that for any classical knot
group there is a virtual knot with that group and trivial longitude. It is well
known that the second homology of a classical knot group is trivial. We provide
counterexamples of this for virtual knots.
  For an arbitrary group G, we give necessary and sufficient conditions for the
existence of a virtual knot group that maps onto G with specified behavior on
the peripheral subgroup. These conditions simplify those that arise in the
classical setting.
