  We present a review of recent advances in the study of many-body effects in
magnetic nanoparticles. Considering classical spins on a lattice coupled by the
exchange interaction in the presence of the bulk and surface anisotropy, we
investigate the effects of finite size, free boundaries, and surface anisotropy
on the average and local magnetization for zero and finite temperatures and
magnetic fields. Superparamagnetism of magnetic particles necessitates
introducing two different, induced and intrinsic, magnetizations. We check the
validity of the much used relation between them within different theoretical
models. We show that the competition between the exchange and surface
anisotropy leads to spin canting dependent on the orientation of the average
magnetization with respect to the crystallographic axes and thus to a
second-order effective anisotropy of the particle. We have also investigated
the switching mechanism of the magnetization upon varying the surface
anisotropy constant. Some cases of more realistic particles are also dealt
with.
