Two operators $T_1,T_2$ on a Hilbert space are called similar if there exists an invertible operator $S$ on $H$ such that $T_1=S^{-1} T_2 S$. A ical problem (the “Halmos problem”) asks for a characterization of the operators $T$ which are similar to a contraction. Two important variants have appeared. One proposed by Dixmier in 1950 asks whether the similarity property for all uniformly bounded representations on a group $G$ is equivalent to the amenability of $G$. The other one, proposed by Kadison in 1995, is about representations of a $C^*$-algebra. This report attempts to describe the genesis and the present state of these problems.