Active acoustic metasurfaces (AAMSs) have been recently recognized as very efficient sound shielding structures, which can have large lateral dimensions perpendicular to the direction of the sound wave propagation but very short lateral dimension along the direction of the sound wavevector. The sound shielding principle of AAMSs is based on control of the specific acoustic impedance (SAI). This is achieved by means of an active tuning of elastic properties of piezoelectric transducers, which, therefore, represent the core element of the AAMSs. Using this approach, it is possible to actively control the acoustic coefficients of transmission and reflection of AAMSs. An important point, which has been recently discovered, is the fact that the great suppression of the transmission coefficient can be achieved in the regime, when the SAI of the AAMS is negative. The function of the AAMS in varying operational conditions or in a wide frequency range, however, put delicate stability conditions on the negative values of SAI. In order to keep the AAMS in the stable operation, a concept of adaptive acoustic metasurfaces (AdAMSs) is introduced in this paper. Methods for the real-time estimation and the active control of the SAI values of the AdAMSs are presented. It is shown that the accurate control of the distribution of the SAI on the surface of the AdAMS enables to control the transmitted sound field not only in the magnitude but also in the direction of the transmitted sound wave.