  I present the junction conditions for F(R) theories of gravity and their
implications: the generalized Israel conditions and equations. These junction
conditions are necessary to construct global models of stars, galaxies, etc.,
where a vacuum region surrounds a finite body in equilibrium, as well as to
describe shells of matter and braneworlds, and they are stricter than in
General Relativity in both cases. For the latter case, I obtain the field
equations for the energy-momentum tensor on the shell/brane, and they turn out
to be, remarkably, the same as in General Relativity. An exceptional case for
quadratic F(R), previously overlooked in the literature, is shown to arise
allowing for a discontinuous R, and leading to an energy-momentum content on
the shell with unexpected properties, such as non-vanishing components normal
to the shell and a new term resembling classical dipole distributions. For the
former case, they do not only require the agreement of the first and second
fundamental forms on both sides of the matching hypersurface, but also that the
scalar curvature R and its first derivative agree there too. I argue that, as a
consequence, matched solutions in General Relativity are not solutions of
F(R)-models generically. Several relevant examples are analyzed.
