We present recent results regarding the characterization of the joint in-
variant subspaces under the universal model $(B_1,\dots, B_n)$ associated with a
noncommutative variety V in a regular domain in $B(H)^n$. The main result
is a Beurling-Lax-Halmos type representation which is used to parameter-
ize the coresponding wandering subspaces of the joint invariant subspaces of
$(B_1\otimes I_E,\ldots, B_n\otimes I_E)$. We also present a characterization of the elements
in the noncommutative variety V which admit characteristic functions. This
leads to an operator model theory for completely non-coisometric elements
which allows us to show that the characteristic function is a complete unitary
invariant for this class of elements. Our results apply, in particular, in the
commutative case.