  We develop the concept of supersymmetry in singular spaces, apply it in an
example for 3-branes in D=5 and comment on 8-branes in D=10. The new
construction has an interpretation that the brane is a sink for the flux and
requires adding to the standard supergravity a (D-1)-form field and a
supersymmetry singlet field. This allows a consistent definition of
supersymmetry on a S_1/Z_2 orbifold, the bulk and the brane actions being
separately supersymmetric. Randall-Sundrum brane-worlds can be reproduced in
this framework without fine tuning. For fixed scalars, the doubling of unbroken
supersymmetries takes place and the negative tension brane can be pushed to
infinity. In more general BPS domain walls with 1/2 of unbroken
supersymmetries, the distance between branes in some cases may be restricted by
the collapsing cycles of the Calabi-Yau manifold. The energy of any static
x^5-dependent bosonic configuration vanishes, E=0, in analogy with the
vanishing of the Hamiltonian in a closed universe.
