Alfvén's derivation of his current limit is given. It demonstrates that it does not give the maximum possible current of a beam, but the maximum current that can propagate for an indefinite distance and time, from a source, in a charge neutral beam. Furthermore, the value Alfvén obtained applies to a uniform current density and to particles initially moving in the direction of the beam. It is also shown that Alfvén predicted that beams which exceed the limit will filament as a result of the particles that are turned back by the magnetic field. His work is extended to beams with particles that have transverse momentum, to beams with non-uniform current densities, to beams that are not charge neutral and to the time dependent case. These extensions of Alfvén's work are found to require numerical calculations in most cases and to give ambiguous results in some cases. A general formula for the current limit is given based on the conservation of energy. It is calculated for the cases considered previously and found to confirm the accuracy of Alfvén's original estimate. The relevance of the current limit to high intensity laser-solid interactions and fast ignition is then discussed.