  In this paper we solve several problems concerning joint similarity to
n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated
with positive regular free holomorphic functions in $n$ noncommuting variables
and with sets of noncommutative polynomials in $n$ indeterminates, where
$B(\cH)$ is the algebra of all bounded linear operators on a Hilbert space
$\cH$. In particular, if $f=X_1+...+X_n$ and $\cP=\{0\}$, the elements of the
corresponding variety can be seen as noncommutative multivariable analogues of
Agler's $m$-hypercontractions.
