The contactless inductive flow tomography (CIFT) aims at reconstructing the velocity field of electrically conducting fluids, with special focus on applications in metallurgy and crystal growth technologies. The method relies on the induction of secondary magnetic fields if the moving fluid is exposed to a primary magnetic field. The theoretical foundation of the method is delineated, and some early experiments on the reconstruction of the three-dimensional flow in a cylinder are sketched. Then, the recent efforts to apply CIFT to various model problems in connection with the continuous casting of steel are summarized.