  This Minireview summarizes the recent efforts to solve forward and inverse
problems as they occur in different branches of fundamental and applied
magnetohydrodynamics. As for the forward problem, the main focus is on the
numerical treatment of induction processes, including self-excitation of
magnetic fields in non-spherical domains and/or under the influence of
non-homogeneous material parameters. As an important application of the
developed numerical schemes, the functioning of the von-K\'{a}rm\'{a}n-sodium
(VKS) dynamo experiment is shown to depend crucially on the presence of
soft-iron impellers. As for the inverse problem, the main focus is on the
mathematical background and some first practical applications of the
Contactless Inductive Flow Tomography (CIFT), in which flow induced magnetic
field perturbations are utilized for the reconstruction of the velocity field.
The promises of CIFT for flow field monitoring in the continuous casting of
steel are substantiated by results obtained at a test rig with a low melting
liquid metal. While CIFT is presently restricted to flows with low magnetic
Reynolds numbers, some selected problems of non-linear inverse dynamo theory,
with possible application to geo- and astrophysics, are also discussed.
