  This paper reviews recent results on hybrid inverse problems, which are also
called coupled-physics inverse problems of multi-wave inverse problems. Inverse
problems tend to be most useful in, e.g., medical and geophysical imaging, when
they combine high contrast with high resolution. In some settings, a single
modality displays either high contrast or high resolution but not both. In
favorable situations, physical effects couple one modality with high contrast
with another modality with high resolution. The mathematical analysis of such
couplings forms the class of hybrid inverse problems.
  Hybrid inverse problems typically involve two steps. In a first step, a
well-posed problem involving the high-resolution low-contrast modality is
solved from knowledge of boundary measurements. In a second step, a
quantitative reconstruction of the parameters of interest is performed from
knowledge of the point-wise, internal, functionals of the parameters
reconstructed during the first step. This paper reviews mathematical techniques
that have been developed in recent years to address the second step.
  Mathematically, many hybrid inverse problems find interpretations in terms of
linear and nonlinear (systems of) equations. In the analysis of such equations,
one often needs to verify that qualitative properties of solutions to elliptic
linear equations are satisfied, for instance the absence of any critical
points. This paper reviews several methods to prove that such qualitative
properties hold, including the method based on the construction of complex
geometric optics solutions.
