We propose a framework called audio inpainting for the general problem of estimating missing samples in audio. It extends the problem of the interpolation and extrapolation of signals to the cases where possibly-large blocks of consecutive samples must be estimated from the remaining, known samples. We relate this framework to a number of applications including declicking, declipping and audio packet loss in voice over IP. By considering audio inpainting as an inverse problem, we show that sparse representations are an appropriate scheme to develop new approaches for the audio inpainting problem. We will present some experiments, with a particular focus on restoration of clipped speech or music signals.