We develop a general procedure for estimating the effective constitutive behavior of nonlinear dielectrics. The procedure is based on a variational principle expressing the effective energy function of a given nonlinear composite in terms of the effective energy functions of the class of linear comparison composites. This provides an automatic procedure for converting well-known information for linear composites, in the form of estimates and bounds for their effective dielectric constants, into corresponding estimates and bounds for the effective behavior of nonlinear composites. Further, the procedure is easily implemented, and leads in some cases to exact results. Thus, exact estimates are given herein for isotropic weakly nonlinear composites with general nonlinearity, and bounds of the Hashin-Shtrikman type are given for the class of two-phase, isotropic dielectric composites with strongly and perfectly nonlinear constitutive behavior. The optimality of the bounds is addressed briefly.