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A driving question in 3-dimensional topology is how the structure of a 3-manifold M with torus boundary can change under Dehn filling. For example, if M contains an essential surface, the surface will usually remain essential after Dehn filling, the exceptions occurring for a restricted set of filling slopes. Is the same true for the entire set of essential surfaces in M? In other words, will this set usually be preserved under Dehn filling except for a restricted set of filling slopes? We will show that the answer is yes. The techniques involved use a kind of thin position argument and some normal surface theory. This is joint work with Dave Bachman and Eric Sedgwick.