This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs and extend it to a more general class of noncommutative polydomains
D
q
m
(
H
)
\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})
in
B
(
H
)
n
B(\mathcal {H})^n
. An important role in our study is played by noncommutative Berezin transforms associated with the elements of the polydomain. These transforms are used to prove that each such polydomain has a universal model
W
=
{
W
i
,
j
}
\mathbf {W}=\{\mathbf {W}_{i,j}\}
consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra
F
∞
(
D
q
m
)
F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})
as the weakly closed algebra generated by
{
W
i
,
j
}
\{\mathbf {W}_{i,j}\}
and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in
D
q
m
(
H
)
\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})
. It is shown that the Berezin transform is a completely isometric isomorphism between
F
∞
(
D
q
m
)
F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})
and the algebra of bounded free holomorphic functions on the radial part of
D
q
m
(
H
)
\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})
. A characterization of the Beurling type joint invariant subspaces under
{
W
i
,
j
}
\{\mathbf {W}_{i,j}\}
is also provided.
It has been an open problem for quite some time to find significant classes of elements in the commutative polydisc for which a theory of characteristic functions and model theory can be developed along the lines of the Sz.-Nagy–Foias theory of contractions. We give a positive answer to this question, in our more general setting, providing a characterization for the class of tuples of operators in
D
q
m
(
H
)
\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})
which admit characteristic functions. The characteristic function is constructed explicitly as an artifact of the noncommutative Berezin kernel associated with the polydomain, and it is proved to be a complete unitary invariant for the class of completely non-coisometric tuples. Using noncommutative Berezin transforms and
C
∗
C^*
-algebra techniques, we develop a dilation theory on the noncommutative polydomain
D
q
m
(
H
)
\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})
.