The ”completely bounded” or ”cb” norm of a linear operator on a C*-algebra, if it is finite, is a much more rigid construct than the usual operator norm since it is obtained via an averaging procedure, i.e. tensoring with the compact operators. As a result, it is often easier to compute and to deal with than the operator norm. In this survey, we shall illustrate this phenomenon by a recent joint contribution together with R. J. Archbold and D. W. B. Somerset to the norm problem for elementary operators on C*-algebras. We will also identify those C*-algebras for which the norm and the cb-norm for every such operator coincide.