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This paper addresses the problem of predicting the quality of telephone speech. Starting from a definition of quality, which takes communicative as well as service-related factors into account, a new classification scheme for prediction models is proposed. It considers input and output parameters, the network components and application area the model is used for, as well as the psychoacoustic and judgment-related bases. According to this scheme, quality prediction models can be classified into signal-based comparative measures, network planning models and monitoring models. Whereas signal-based approaches have been described extensively in literature, this paper discusses the latter two approaches in detail. The underlying psychoacoustic properties of two network planning models, the E-model and the SUBMOD model, are analyzed, and combined approaches for monitoring models are developed. Quality predictions obtained from the models are compared to the results of auditory test data, and weaknesses as well as network elements that remain uncovered are identified. Possible future extensions to the models are pointed out, including wide-band scenarios and speech sound quality, non-stationary impairments as well as speech technology devices.