We analyze in detail the possible breaking of space-time supersymmetry under T-duality transformations. We find that when appropriate world-sheet effects are taken into account apparent puzzles concerning supersymmetry in space-time are solved. We study T-duality in general heterotic σ-models analyzing possible anomalies, and we find some modifications of Buscher's rules. We then work out a simple but representative example which contains most of the difficulties. We also consider the issue of supersymmetry versus duality for marginal deformations of WZW models and present a mechanism that restores supersymmetry dynamically in the effective theory.