  In this paper, we study noncommutative varieties in polydomains in $B(H)^n$.
The goal is to understand the structure of these varieties, determine their
elements and classify them up to unitary equivalence. Using noncommutative
Berezin transforms associated with each variety, we develop an operator model
theory and dilation theory for large classes of varieties in noncommutative
polydomains. This includes various commutative cases which are close connected
to the theory of holomorphic functions in several complex variables and
algebraic geometry.
