The electronic and topological properties of alpha-Sn (a-Sn) and HgTe and their low-dimensional systems are studied by means of ab-initio calculations. The used approximate quasiparticle theory including spin-orbit coupling (SOC) gives the correct band ordering, band gap and SOC-splitting for bulk a-Sn, HgTe and CdTe. Calculating the topological invariants, a-Sn and HgTe are shown to be topological insulators, whereas CdTe is identified as trivial insulator. These findings suggest the occurrence of the quantum-spin Hall (QSH) phase in the respective low-dimensional systems, which is characterized by topologically protected spin-polarized edge-of surface states. For a-Sn nanocrystals, a topological transition indicated by an inversion of the electronic level-ordering for a certain critical diameter is found. In addition, the diameter-dependent optical absorption-and emission properties are investigated in detail. In a-Sn(001) surfaces, the occurrence of spin-polarized topological surface states, although inside the inverted bulk band gap is demonstrated in agreement to recent angle-resolved photoemission studies. The investigation of HgTe/CdTe and a-Sn/CdTe quantum-well (QW) structures shows a topological transition from the topological trivial insulating state into a non-trivial QSH insulator with increasing QW thickness. The spatial localization of the wave functions near the interfaces and the spin polarization are demonstrated for the edge states for QWs with thicknesses near the critical one. They do not depend on the QW orientation and are therefore topologically protected. We show that the inclusion of inversion symmetry, the non-axial rotation symmetry of the QWs, and the real QW barriers lead to some agreement but also significant deviations from the predictions within toy models. Real structure effects becom most improtant in a-Sn/CdTe(001) Qws, where the QSH phase is suppressed by an intrinsic electrostatic saw-tooth potential.