  We study generalized multifractality characterizing fluctuations and
correlations of eigenstates in disordered systems of symmetry classes AII, D,
and DIII. Both metallic phases and Andersonlocalization transitions are
considered. By using the non-linear sigma-model approach, we construct
pure-scaling eigenfunction observables. The construction is verified by
numerical simulations of appropriate microscopic models, which also yield
numerical values of the corresponding exponents. In the metallic phases, the
numerically obtained exponents satisfy Weyl symmetry relations as well as
generalized parabolicity (proportionality to eigenvalues of the quadratic
Casimir operator). At the same time, the generalized parabolicity is strongly
violated at critical points of metal-insulator transitions, signalling
violation of local conformal invariance. Moreover, in classes D and DIII, even
the Weyl symmetry breaks down at critical points of metal-insulator
transitions. This last feature is related with a peculiarity of the sigma-model
manifolds in these symmetry classes: they consist of two disjoint components.
Domain walls associated with these additional degrees of freedom are crucial
for ensuring Anderson localization and, at the same time, lead to the violation
of the Weyl symmetry.
