The nature of the interfaces between SiGe nanostructures and Si in heterostructures strongly affects carrier mobility and recombination for physical confinement in one, two, and three dimensions. The interface sharpness is influenced by many factors including growth conditions, strain, and thermal processing, which can make it difficult to attain the desired structures. This is certainly the case for nanostructure confinement in one dimension. However, axial Si/Ge nanowire heterojunctions with a Si/Ge nanowire diameter in the range 50-120 nm produce a strong photoluminescence signal associated with band-to-band electron-hole recombination at the nanowire heterojunction that is attributed to a specific interfacial SiGe alloy composition. For three-dimensional confinement, experiments show that two quite different SiGe nanostructures incorporated into a Si0.6Ge0.4 wavy structure exhibit an intense PL signal with a characteristic non-exponential decay time that is remarkably shorter (as much as 1000 times) than that found in conventional Si/SiGe nanostructures.