We show that the set of closed, essential, 2-sided surfaces (considered up to isotopy) in a 3-manifold with a torus boundary component survives unchanged in all suitably generic Dehn fillings. Furthermore, for all but finitely many non-generic fillings, we show that two essential surfaces can only become isotopic in a very constrained way. If time permits, we will also sketch future work on the persistence of the set of Heegaard surfaces after generic Dehn filling. This is joint work with Ryan Derby-Talbot and Eric Sedgwick.