The magnetic impurity problem is that transition metal impurities are magnetic in some, but not all, simple metal solvents. There is a whole spectrum of anomalous effects associated with the occurrence of magnetism (as defined in Section 2), particularly in the temperature variation of the low temperature resistivity, the size of the thermo power, and a drastic effect on the superconductivity of the solvent. The magnetic impurity problem became the Kondo problem when Kondo in 1964 explained these anomalies in terms of repeated spin flip scattering between the impurity and the host conduction electrons. It was soon realized that the presence or absence of magnetism was itself intimately tied up with the multiple spin scattering, and that the division between magnetic and non-magnetic is not as sharp as had been thought. The problem has now achieved a certain unity, largely because careful experiments have shown that physically these systems are always well-behaved, but its rigorous theoretical resolution still remains as a difficult task.