A fusion-fission (hybrid) reactor is a combination of a fusion device and a subcritical fission reactor, where the fusion device acts as a neutron source and the power is mainly produced in the fission core. Hybrid reactors may be suitable for transmutation of transuranic isotopes in the spent nuclear fuel, due to the safety margin on criticality imposed by the subcritical fission core. The SFLM Hybrid project is a theoretical project that aims to point out the possibilities with steady-state mirror-based hybrid reactors. The quadrupolar magnetic mirror vacuum field is based on the Straight Field Line Mirror field and the central cell is 25 m long. A fission mantle surrounds the mirror cell. The fission to fusion power ratio is about 150 with keff = 0.97, implying that almost all the produced energy comes from fission. Beyond each mirror end magnetic expanders are located, which increase the plasma receiving “divertor” area and provide tolerable heat load on wall materials. The plasma is heated with ion cyclotron radio frequency heating and the fission mantle is cooled using a liquid lead-bismuth eutectic. The device is self-sufficient in tritium, and does not seem to suffer from severe material problems. A remaining issue may be the plasma electron temperature, which need to reach about 500 eV for efficient power production.  In this doctoral thesis, theoretical work has been done with the magnetic coil system of such a device and also with the overall concept. A new coil type, the fishbone coil, suitable for single cell quadrupolar mirrors, has been invented. Two vacuum field coil sets with satisfying properties have been found, where the most recent coil set consists of fishbone coils. Finite ß effects on the magnetic field have been investigated, showing that the flux tube ellipticity increases with ß. The ellipticity of the vacuum field increases slightly with radius, but with finite ß it decreases with radius. The maximum flux surface radial extensions decrease with ß, which is an unexpected and beneficial result. A radial invariant has also been identified, and particle simulations have been made to emphasize that quadrupolar mirrors must be symmetric or confinement may be lost.