This concise paper investigates various aspects of the application of unbiased linear minimum mean-square error (ULMMSE) estimators to the equalization of channels used for digital data transmission. One application is to the equalization of the channel to a response free from intersymbol interference. Previous results regarding this application are clarified and extended. The other application is to the equalization of the channel to a finite short memory response. This stems from an interest in reducing the complexity of the Viterbi algorithm. Various new results on the optimum ULMMSE equalizer and its performance are presented. In both applications, the optimum ULMMSE equalizer is a stable realizable, finite dimensional recursive filter.