  Metallic ferromagnetism is in general an intermediate to strong coupling
phenomenon. Since there do not exist systematic analytic methods to investigate
such types of problems, the microscopic origin of metallic ferromagnetism is
still not sufficiently understood. However, during the last two or three years
remarkable progress was made in this field: It is now certain that even in the
one-band Hubbard model metallic ferromagnetism is stable in dimensions $d=1,$
2, and $\infty$ on regular lattices and at intermediate values of the
interaction $U$ and density $n$. In this paper the basic questions and recent
insights regarding the microscopic conditions favoring metallic ferromagnetism
in this model are reviewed. These findings are contrasted with the results for
the orbitally degenerate case.
