First, several impedance spectroscopy models found useful for data fitting and interpretation of dielectric- and conductive-system frequency data are briefly summarized and compared. The first of these are the Kohlrausch KD model for dielectric-system response, representing the frequency response associated with stretched-exponential temporal response, and models appropriate for ionically conducting materials. These are the K0 model for fitting and analysis (equivalent to the KD model but applied to conducting situations), and its transform extensions: the original modulus formalism and the important corrected modulus formalism, both involving the basic K1 model, calculated from the K0 model. Also, several approaches to describing and fitting two types of nearly-constant-loss effects in conductive-systems are compared. Finally, previously unpublished comparisons and evaluations of several coupling models and the cutoff model are presented and show that the results of the coupling models are inconsistent and lead to physically implausible high-temperature response. The cutoff model does not suffer from these deficiencies, and, as well, its physical basis is simpler than that of the coupling models. It is concluded, on the basis of both experimental and synthetic data fitting, that for conductive-system situations the corrected modulus formalism approach, with a temperature- and ionic-concentration-independent fractional exponent, beta(1C)congruent to 1/3, is the most appropriate model for representing the hopping response of homogeneous materials; an effective-medium nearly-constant-loss model describes such experimental loss data well; and the cutoff model, appropriate in both the time and the frequency domains, should replace all coupling models. (c) 2005 Elsevier B.V. All rights reserved.