  The Kondo effect may develop in those cases where there are non-commuting
operators describing the interaction between the conduction electrons and
impurities or defects with internal degrees of freedom. This interaction may
involve spin or orbital variables. There are cases where the conduction
electrons have conserved quantum numbers that do not appear in the coupling. An
example is where the Kondo effect involves orbital degrees of freedom, and the
interaction is independent of the real spins of the electrons, which are
conserved and lead to the two-channel Kondo (2CK) problem. The low temperature
behavior is very different for the one- and two-channel cases, as in the first
case a Fermi liquid is formed while in the second one strong deviations appear
and a non-Fermi liquid state is realized. Mesoscopic samples provide a unique
possibility to study a few or even a single Kondo impurities. In this paper we
first review how the original spin Kondo problem is affected by surface
anisotropy and the fluctuations of the density of states in point contacts
(PCs). We discuss the physics of a single magnetic impurity on the surface of a
sample. The orbital 2CK effect due to dynamical defects is considered. The
nature of these defects is not known but they are excellent candidates to
describe the zero-bias anomalies with non-Fermi liquid character in PCs, and
the dephasing time and transport in short wires. There are two main concerns
regarding this interpretation: First, the tunneling centers formed by heavy
impurities may produce a Kondo temperature that is too low, and secondly, the
splittings seen in the experiments are much smaller than expected from this
model. It would be therefore extremely important to identify the microstructure
of these two-level systems and find new realizations for them.
