Kadison and Kastler introduced a metric on the set of all C$^*$-algebras on a
fixed Hilbert space. In this paper structural properties of C$^*$-algebras
which are close in this metric are examined. Our main result is that the
property of having a positive answer to Kadison's similarity problem transfers
to close C$^*$-algebras. In establishing this result we answer questions about
closeness of commutants and tensor products when one algebra satisfies the
similarity property. We also examine $K$-theory and traces of close
C$^*$-algebras, showing that sufficiently close algebras have isomorphic
Elliott invariants when one algebra has the similarity property.