  We develop a dilation theory for row contractions subject to constraints
determined by sets of noncommutative polynomials. Under natural conditions on
the constraints, we have uniqueness for the minimal dilation. A characteristic
function is associated with any (constrained) row contraction. For pure
constrained row contraction, we show that the characteristic function is a
complete unitary invariant and provide a model. We also show that the curvature
invariant and Euler characteristic asssociated with a Hilbert module generated
by an arbitrary (resp. commuting) row contraction can be expressed only in
terms of the corresponding (resp. constrained) characteristic function. We
provide a commutant lifting theorem for pure constrained row contractions and
obtain a Nevanlinna-Pick interpolation result in this setting.
  Our theory includes, in particular, the commutative and q-commutative cases,
while the standard noncommutative dilation theory for row contractions serves
as a "universal model".
