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This volume extends the ISSI series on magnetic fields in the Universe into the domain of what are by far the strongest fields in the Universe, and stronger than any field that could be produced on Earth. The chapters describe the magnetic fields in non-degenerate strongly magnetized stars,in degenerate stars (such as white dwarfs and neutron stars), exotic members called magnetars, and in their environments, as well as magnetic fields in the environments of black holes. These strong fields have a profound effect on the behavior of matter, visible in particular in highly variable processes like radiation in all known wavelengths, including Gamma-Ray bursts. The generation and structure of such strong magnetic fields and effects on the environment are also described.