  We obtain intertwining dilation theorems for noncommutative regular domains
D_f and noncommutative varieties V_J of n-tuples of operators, which generalize
Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting
contractions. We present several applications including a new proof for the
commutant lifting theorem for pure elements in the domain D_f (resp. variety
V_J) as well as a Schur type representation for the unit ball of the Hardy
algebra associated with the variety V_J. We provide Ando type dilations and
inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V_I. In
particular, we obtain extensions of Ando's results and Agler-McCarthy's
inequality for commuting contractions to larger classes of commuting operators.
