Not yet reviewed by Pith; the record is open.

Faraday tomography is a new method of the study of cosmic magnetic fields enabled by broadband low-frequency radio observations. By Faraday tomography, it is possible to obtain the Faraday dispersion function which contains information on the line-of-sight distributions of magnetic fields, thermal electron density, and cosmic-ray electron density by measuring the polarization spectrum from a source of synchrotron radiation over a wide band. Furthermore, by combining it with 2-dimensional imaging, Faraday tomography allows us to explore the 3-dimensional structure of polarization sources. The application of Faraday tomography has been active in the last 20 years, when broadband observation has become technically feasible. However, the Faraday dispersion function is mathematically the Fourier transform of the polarization spectrum, and since the observable band is finite, it is impossible to obtain a complete Faraday dispersion function by performing Fourier transform. In addition, the Faraday dispersion function does not directly reflect the distribution of magnetic field, thermal-electron density, and cosmic-ray electron density in the physical space, and its physical interpretation is not straightforward. Despite these two difficult problems, Faraday tomography is attracting much attention because it has great potential as a new method for studying cosmic magnetic fields and magnetized plasmas. In particular, the next-generation radio telescope SKA (Square Kilometre Array) is capable of polarization observation with unprecedented sensitivity and broad bands, and the application of Faraday tomography is expected to make dramatic progress in the field of cosmic magnetic fields. In this review, we explain the basics of Faraday tomography with simple and instructive examples. Then representative algorithms to realize Faraday tomography are introduced and finally some applications are shown.