The acoustic and dielectric properties of different glasses at audio frequencies and temperatures below 1 K have been investigated with the vibrating reed and a capacitance bridge technique. We found the temperature dependence of the absorption of vitreous silica (Suprasil W) to agree with the predictions of the tunneling model which is commonly used to explain the low temperature behaviour of amorphous materials. The variation of the sound velocity and of the dielectric constant, however, shows significant deviations from the expected behaviour which cannot be accounted for by a simple modification of the model. Instead, it seems to be necessary to introduce a temperature dependence of some relevant model parameters. Moreover, at very low temperatures (T < 0.1 K) the sound velocity strongly depends on the excitation levels. The absence of this effect at higher temperatures proves that it can be ascribed to a nonlinear response of tunneling systems. Similar results were found in sound velocity measurements on a cover glass and on a superconducting metallic glass (Pd30Zr70, Tc = 2.6 K), which indicates that these features are a general aspect of the dynamics of tunneling states in glasses. In contrast to the insulating glasses we found that in Pd30Zr70 also the internal friction is strain dependent.