We will survey some recent progress on certain similarity problems which can be posed in various contexts. In the group case the problem has a long history going back to Sz. Nagy and Dixmier and may be formulated as: on which groups G is every uniformly bounded representation p : G Æ B(H) "unitarizable" (i.e. similar to a unitary one)? For the disc algebra, the analogous problem was posed by Halmos in 1970 and was recently solved: there is a polynomially bounded operator which is not similar to a contraction. For C*-algebras, the corresponding problem (posed in 1955 by Kadison) is still open: is every bounded homomorphism on a C*-algebra similar to a contractive one? The more recent concept of a "completely bounded map" allows us to treat these problems in a common framework. In each case (either groups, uniform algebras, or C*-algebras), the appropriate notion of "amenability" plays an important role in the recent developments that will be presented.
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