Supersymmetry in singular spaces

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 co nstruction 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 sup ersymmetry singlet field. This allows a consistent definition of supersymme try 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 fra mework without fine tuning. For fixed scalars, the doubling of unbroken sup ersymmetries takes place and the negative tension brane can be pushed to in finity. In more general BPS domain walls with 1/2 of unbroken supersymmetri es, the distance between branes in some cases may be restricted by the coll apsing cycles of the Calabi-Yau manifold. The energy of any static x(5)-dep endent bosonic configuration vanishes, E = 0, in analogy with the vanishing of the hamiltonian in a closed universe.