  Several examples of (separable) C*-algebras with the property that their
second (iterated) local multiplier algebra is strictly larger than the first
have been found by various groups of authors over the past few years, thus
answering a question originally posed by G. K. Pedersen in 1978. This survey
discusses a systematic approach by P. Ara and the author to produce such
examples on the one hand; on the other hand, we present new criteria
guaranteeing that the second and the first local multiplier algebra of a
separable C*-algebra agree. For this class of C*-algebras, each derivation of
the local multiplier algebra is inner.
