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.