The impedance of a brass instrument has an important influence on its playability and sound timbre. The geometry of the mouthpiece has various features, such as the cup volume and shape, opening diameter, and length, that determine the characteristics of the overall impedance of the instrument-mouthpiece combination. Brass instruments, and especially mouthpieces, are designed for specific purposes, and horns or mouthpieces are chosen depending on the musical requirements. In order to investigate the relationship between the physical parameters and the impedances of instruments and mouthpieces, they have been modeled with transfer matrix and finite element model techniques, and the results are compared with impedance measurements of instruments, mouthpieces, and combinations of instruments and mouthpieces. Trumpets, flugelhorns, (French) horns, trombones, and the corresponding mouthpieces have been used. A detailed analysis of the estimation of the viscothermal losses has been performed, as the loss estimation in the narrow throat of the mouthpiece and in the flaring part of the brass instrument bell departs from the ordinary transfer matrix calculations. The effect of varying the physical parameters of mouthpieces and instruments is investigated by means of impedance considerations and sound synthesis, and the resulting influences on intonation, playability, and timbre are presented.