The unification problem for terms containing associative and commutative functions is of importance in theorem provers based on term rewriting and resolution methods as well as in logic programming. The complexity of determining whether two such terms are unifiable was known to be NP-hard. It is proved that the problem is NP-complete by describing a nondeterministic polynomial time algorithm for it. The case where the terms are linear and have no common variables is shown to be in P. The NP-completeness of other similar unification problems, in particular, when a function symbol is also idempotent and/or has a unit (identity), is also discussed. Finally, a table of the complexity of E-matching and E-unification problems is given.