As an alternative to the fast ignition by petawatt lasers of small spherical deuterium–tritium (DT) targets compressed to thousand times solid density, the fast ignition by terawatt relativistic electron beams of thin cylindrical (or conical) DT targets, compressed to several ten times solid density and magnetized to 108 G through a high current discharge along the cylindrical axis of the targets, has been proposed. One problem of this approach is the guiding and focusing of the relativistic electron beam onto the target. It is proposed to transport the beam through a laser-triggered ionized channel in a low density background gas, or by letting it propagate along the surface of a thin wire crossing the diode gap, with the final focusing done by repulsive image currents in a conducting convergent cone. A second problem is the stopping of the electron beam in the target. This can hopefully be done by a combination of classical electron stopping power, the electrostatic two-stream instability and collisionless shocks in the presence of a strong perpendicular magnetic field.