  We study the maximally supersymmetric Kondo model obtained by adding a
fermionic impurity to N=4 supersymmetric Yang-Mills theory. While the original
Kondo problem describes a defect interacting with a free Fermi liquid of
itinerant electrons, here the ambient theory is an interacting CFT, and this
introduces qualitatively new features into the system. The model arises in
string theory by considering the intersection of a stack of M D5-branes with a
stack of N D3-branes, at a point in the D3 worldvolume. We analyze the theory
holographically, and propose a dictionary between the Kondo problem and
antisymmetric Wilson loops in N=4 SYM. We perform an explicit calculation of
the D5 fluctuations in the D3 geometry and determine the spectrum of defect
operators. This establishes the stability of the Kondo fixed point together
with its basic thermodynamic properties. Known supergravity solutions for
Wilson loops allow us to go beyond the probe approximation: the D5s disappear
and are replaced by three-form flux piercing a new topologically non-trivial
three-sphere in the corrected geometry. This describes the Kondo model in terms
of a geometric transition. A dual matrix model reflects the basic properties of
the corrected gravity solution in its eigenvalue distribution.
