  Recently, there has been a resurgence of intense experimental and theoretical
interest on the Kondo physics of nanoscopic and mesoscopic systems due to the
possibility of making experiments in extremely small samples. We have carried
out exact diagonalization calculations to study the effect of energy spacing
$\Delta$ in the conduction band states, hybridization, number of electrons, and
disorder on the ground-state and thermal properties of strongly-correlated
electron nanoclusters. For the ordered systems, the calculations reveal for the
first time that $\Delta$ tunes the interplay between the {\it local} Kondo and
{\it non local} RKKY interactions, giving rise to a "Doniach phase diagram" for
the nanocluster with regions of prevailing Kondo or RKKY correlations. The
interplay of $\Delta$ and disorder gives rise to a $\Delta$ versus
concentration T=0 phase diagram very rich in structure. The parity of the total
number of electrons alters the competition between the Kondo and RKKY
correlations. The local Kondo temperatures, $T_K$, and RKKY interactions depend
strongly on the local environment and are overall {\it enhanced} by disorder,
in contrast to the hypothesis of ``Kondo disorder'' single-impurity models.
This interplay may be relevant to experimental realizations of small rings or
quantum dots with tunable magnetic properties.
