  We define a class of boundary value problems on manifolds with fibered
boundary. This class is in a certain sense a deformation between the classical
boundary value problems and the Atiyah-Patodi-Singer problems in subspaces. The
boundary conditions in this theory are taken as elements of the C^*-algebra
generated by pseudodifferential operators and families of pseudodifferential
operators in the fibers. We prove the Fredholm property for elliptic boundary
value problems. This class also contains nonlocal boundary value problems of
math.KT/0108107. We compute a topological obstruction (similar to the
Atiyah-Bott obstruction) to the existence of elliptic boundary conditions for a
given operator. Geometric operators with a nontrivial obstruction are given.
