Statistical ill-posed inverse problems are becoming increasingly important in a diverse range of disciplines, including geophysics, astronomy, medicine and economics. Roughly speaking, in all of these applications the observable signal g=Tf is a transformation of a functional parameter of interest f under a transformation T. Statistical inference on f based on an estimation of g which usually necessitates an inversion of T is thus called an inverse problem. Moreover, by ill-posed we mean that the transformation T is not stable, i.e., T has not a continuous inverse. In most applications, however, both the signal g and the inherent transformation T are not known in practice, although they can be estimated from the data. Consequently, a statistical inference has to take into account that a random noise is present in both the estimated signal and the estimated operator.