  Conventional Magnetic Resonance Imaging (MRI) is hampered by long scan times
and only qualitative image contrasts that prohibit a direct comparison between
different systems. To address these limitations, model-based reconstructions
explicitly model the physical laws that govern the MRI signal generation. By
formulating image reconstruction as an inverse problem, quantitative maps of
the underlying physical parameters can then be extracted directly from
efficiently acquired k-space signals without intermediate image reconstruction
-- addressing both shortcomings of conventional MRI at the same time. This
review will discuss basic concepts of model-based reconstructions and report
about our experience in developing several model-based methods over the last
decade using selected examples that are provided complete with data and code.
