  A nano-scale metallic grain (nanoparticle) with irregular boundaries in which
the single-particle dynamics are chaotic is a zero-dimensional system described
by the so-called universal Hamiltonian in the limit of a large number of
electrons. The interaction part of this Hamiltonian includes a superconducting
pairing term and a ferromagnetic exchange term. Spin-orbit scattering breaks
spin symmetry and suppresses the exchange interaction term. Of particular
interest is the fluctuation-dominated regime, typical of the smallest grains in
the experiments, in which the bulk pairing gap is comparable to or smaller than
the single-particle mean-level spacing, and the Bardeen-Cooper-Schrieffer (BCS)
mean-field theory of superconductivity is no longer valid. Here we study the
crossover between the BCS and fluctuation-dominated regimes in two limits. In
the absence of spin-orbit scattering, the pairing and exchange interaction
terms compete with each other. We describe the signatures of this competition
in thermodynamic observables, the heat capacity and spin susceptibility. In the
presence of strong spin-orbit scattering, the exchange interaction term can be
ignored. We discuss how the magnetic-field response of discrete energy levels
in such a nanoparticle is affected by pairing correlations. We identify
signatures of pairing correlations in this response, which are detectable even
in the fluctuation-dominated regime.
