  A 3-manifold is Haken if it contains a topologically essential surface. The
Virtual Haken Conjecture says that every irreducible 3-manifold with infinite
fundamental group has a finite cover which is Haken. Here, we discuss two
interrelated topics concerning this conjecture.
  First, we describe computer experiments which give strong evidence that the
Virtual Haken Conjecture is true for hyperbolic 3-manifolds. We took the
complete Hodgson-Weeks census of 10,986 small-volume closed hyperbolic
3-manifolds, and for each of them found finite covers which are Haken. There
are interesting and unexplained patterns in the data which may lead to a better
understanding of this problem.
  Second, we discuss a method for transferring the virtual Haken property under
Dehn filling. In particular, we show that if a 3-manifold with torus boundary
has a Seifert fibered Dehn filling with hyperbolic base orbifold, then most of
the Dehn filled manifolds are virtually Haken. We use this to show that every
non-trivial Dehn surgery on the figure-8 knot is virtually Haken.
