  Degeneracy loci of morphisms between vector bundles have been used in a wide
variety of situations. We introduce a vast generalization of this notion, based
on orbit closures of algebraic groups in their linear representations. A
preferred class of our orbital degeneracy loci is characterized by a certain
crepancy condition on the orbit closure, that allows to get some control on the
canonical sheaf. This condition is fulfilled for Richardson nilpotent orbits,
and also for partially decomposable skew-symmetric three-forms in six
variables. In order to illustrate the efficiency and flexibility of our
methods, we construct in both situations many Calabi--Yau manifolds of
dimension three and four, as well as a few Fano varieties, including some new
Fano fourfolds.
