Reconstructing images from indirect measurements is a central problem in many applications, including the subject of this special issue, quantitative susceptibility mapping (QSM). The process of image reconstruction typically requires solving an inverse problem that is ill-posed and large-scale and thus challenging to solve. Although the research field of inverse problems is thriving and very active with diverse applications, in this part of the special issue we will focus on recent advances in inverse problems that are specific to deconvolution problems, the class of problems to which QSM belongs. We will describe analytic tools that can be used to investigate underlying ill-posedness and apply them to the QSM reconstruction problem and the related extensively studied image deblurring problem. We will discuss state-of-the-art computational tools and methods for image reconstruction, including regularization approaches and regularization parameter selection methods. We finish by outlining some of the current trends and future challenges.