  Inverse problems arise in a variety of imaging applications including
computed tomography, non-destructive testing, and remote sensing. The
characteristic features of inverse problems are the non-uniqueness and
instability of their solutions. Therefore, any reasonable solution method
requires the use of regularization tools that select specific solutions and at
the same time stabilize the inversion process. Recently, data-driven methods
using deep learning techniques and neural networks demonstrated to
significantly outperform classical solution methods for inverse problems. In
this chapter, we give an overview of inverse problems and demonstrate the
necessity of regularization concepts for their solution. We show that neural
networks can be used for the data-driven solution of inverse problems and
review existing deep learning methods for inverse problems. In particular, we
view these deep learning methods from the perspective of regularization theory,
the mathematical foundation of stable solution methods for inverse problems.
This chapter is more than just a review as many of the presented theoretical
results extend existing ones.
