  We consider inverse problems related to the velocity reconstruction in
electrically conducting fluids from externally measured magnetic fields. The
underlying theory is presented in the framework of the integral equation
approach to homogeneous dynamos in finite domains, which can be cast into a
linear inverse problem in case that the magnetic Reynolds number is not too
large. Some mathematical problems of the inversion, including the uniqueness
problem in the sphere and a paradigmatic isospectrality problem for mean-field
dynamos, are touched upon. For practical purposes, the inversion is carried out
with the help of Tikhonov regularization using a quadratic functional of the
velocity field as penalty function. For the first time, we present results of
an experiment in which the three-dimensional velocity field of a propellor
driven flow in a liquid metal is reconstructed by a contactless inductive
measuring technique.
