  Since its first appearance in 1971, Gutzwiller's trace formula has been
extended to systems with continuous symmetries, in which not all periodic
orbits are isolated. In order to avoid the divergences occurring in connection
with symmetry breaking and orbit bifurcations (characteristic of systems with
mixed classical dynamics), special uniform approximations have been developed.
We first summarize some of the recent developments in this direction. Then we
present applications of the extended trace formulae to describe prominent
gross-shell effects of various finite fermion systems (atomic nuclei, metal
clusters, and a mesoscopic device) in terms of the leading periodic orbits of
their suitably modeled classical mean-field Hamiltonians.
