  Regularization methods are a key tool in the solution of inverse problems.
They are used to introduce prior knowledge and make the approximation of
ill-posed (pseudo-)inverses feasible. In the last two decades interest has
shifted from linear towards nonlinear regularization methods even for linear
inverse problems. The aim of this paper is to provide a reasonably
comprehensive overview of this development towards modern nonlinear
regularization methods, including their analysis, applications, and issues for
future research.
  In particular we will discuss variational methods and techniques derived from
those, since they have attracted particular interest in the last years and link
to other fields like image processing and compressed sensing. We further point
to developments related to statistical inverse problems, multiscale
decompositions, and learning theory.
