The Extrap concept and its possibilities as a full-scale fusion reactor are reviewed. The toroidal Extrap configuration consists of a Z-pinch that is immersed in an octupole field generated by currents in a set of ring-shaped external conductors. This configuration satisfies the equilibrium conditions of an optimized compact fusion reactor in having closed field lines, fully axisymmetric geometry, a weak or nonexisting toroidal magnetic field, no need for a surrounding conducting wall, larger bootstrap currents than those in schemes with a dominating toroidal magnetic field, the possible option of normally conducting coils, and a high-beta value. Small- and medium-scale linear and toroidal experiments have demonstrated macroscopic stability at plasma temperatures and poloidal beta values of at least 40 eV and 60%, for electron densities of ∼1021 m−3, discharge durations of the order of 100 Alfvén times, and energy confinement times of ∼40 Alfvén times. The energy confinement time is almost two orders of magnitude longer than the growth times of the most violent magnetohydrodynamic (MHD) instabilities, and the Lawson parameter is ∼1.5 × 1016 s/m3. The stability appears to be explained by a combination of MHD-like and kinetic effects. However, further advanced theoretical methods, partly including unexplored areas, have to be employed in the search for a complete understanding of the experiments. An extrapolation to a full-scale reactor appears to be possible, but requires further investigation. Crucial parameters f or stability are the number θi, of ion Larmor radii contained within the pinch radius and the ratio of the magnetic field strengths generated by the pinch and the conductor currents. In the experiments, θi, ≲ 10, whereas the range 20 ≲ θi ≲ 40 is required for a reactor.