The computably enumerable (c.e.) sets form a lattice, E, in the language of set inclusion. Much work has been done to build automorphisms of E, but there are still many open questions in this area. One such question is the problem of determining which sets can be taken by automorphisms of E to sets of certain degrees. For instance, which sets are automorphic to low sets? We can also ask which sets are effectively automorphic to the low sets. We will look at the history of automorphism constructions as well as discuss recent results in the area, including joint work with Cholak on classifying the sets automorphic to low_2 and low sets.