We show that \most" Dehn-llings of a non-bered, atoroidal, Haken three-manifold with torus boundary are virtually Haken. 1 Results Suppose that X is a compact, oriented, three-manifold with boundary a torus T: We will pick a basis of H 1 (T) represented by simple loops ; such that = 0 in H 1 (X; Q): We call a longitude and a meridian. A slope, ; on T is the isotopy class of essential unoriented...