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This paper reviews recent results on hybrid inverse problems, which are alsocalled coupled-physics inverse problems of multi-wave inverse problems. Inverseproblems tend to be most useful in, e.g., medical and geophysical imaging, whenthey combine high contrast with high resolution. In some settings, a singlemodality displays either high contrast or high resolution but not both. Infavorable situations, physical effects couple one modality with high contrastwith another modality with high resolution. The mathematical analysis of suchcouplings forms the class of hybrid inverse problems.  Hybrid inverse problems typically involve two steps. In a first step, awell-posed problem involving the high-resolution low-contrast modality issolved from knowledge of boundary measurements. In a second step, aquantitative reconstruction of the parameters of interest is performed fromknowledge of the point-wise, internal, functionals of the parametersreconstructed during the first step. This paper reviews mathematical techniquesthat have been developed in recent years to address the second step.  Mathematically, many hybrid inverse problems find interpretations in terms oflinear and nonlinear (systems of) equations. In the analysis of such equations,one often needs to verify that qualitative properties of solutions to ellipticlinear equations are satisfied, for instance the absence of any criticalpoints. This paper reviews several methods to prove that such qualitativeproperties hold, including the method based on the construction of complexgeometric optics solutions.

Hybrid inverse problems and internal functionals