We review analytical and numerical results derived from the Bethe ansa
tz solution of the n-channel Kondo model of arbitrary spin S as a func
tion of temperature, external field, impurity spin S and the number of
channels n. Three situations have to be distinguished: (i) If n=2S th
e conduction electrons exactly compensate the impurity spin into a sin
glet at low temperatures, (ii) if n < 2S the impurity spin is only par
tially compensated (undercompensated), and (iii) if n > 2S the impurit
y spin is said to be overcompensated giving rise to critical behaviour
. The results are discussed in the context of magnetic impurities, e.g
. Fe, Cr and Tm in simple metals, the quadrupolar Kondo effect. an imp
urity spin embedded in the Takhtajan-Babujian Heisenberg model and ele
ctron assisted-tunnelling of an atom in a double-well potential.