This paper is concerned with E-unification in arbitrary equational theories. We extend the method of transformations on systems of terms, developed by Martelli-Montanari for standard unification, to E-unification by giving two sets of transformation, BJ and J , which are proved to be sound and complete in the sense that a complete set of E-unifiers for any equational theory E can be enumerated by either of these sets. The set J is an inprovement of BJ , in that many E-unifiers produced by BJ will be weeded out by J . In addition, we show that a generalization of surreduction (also called narrowing) combined with the computation of critical pairs is complete. A new representation of equational proofs as certain kinds of trees is used to prove the completeness of the set BJ in a rather direct fashion that parallels the completeness of the transformations in the case of (standard) unification. The completeness of J and the generalization of surreduction is proved by a method inspired by the concept of unfailing completion, using an abstract (and simpler) notion of the completion of a set of equations.