Silicon-based heterostructures have come a long way from the discovery of strain as a new and essential parameter for band structure engineering to the present state of electron and hole mobilities, which surpass those achieved in the traditional material combination by more than an order of magnitude and are rapidly approaching the best III - V heteromaterials. It is the purpose of this article to report on the most recent developments, and the performance level achieved to date in this material system, in a concise and critical manner. The first part of this review is concerned with the structural and electronic properties of the lattice-mismatched Si/SiGe heterostructure. Emphases are put on the effects of strain both on the band structure and on the band offsets, as well as on means to actually control the strain in a stack of heteroepitaxial layers. The second part is dedicated to the transport properties of low-dimensional carrier systems in Si/SiGe and Ge/SiGe heterostructures. The prospects and limitations of the different layer concepts are discussed in terms of scattering mechanisms and experimental results. This part also reviews the most recent magneto-transport experiments on quantum wires and quantum point contacts, which became possible by the enhanced mean free paths in these materials. The third part covers the device aspects of these high-mobility materials, which is of special interest, because silicon-based heterostructures can significantly enhance the performance level of contemporary Si devices without sacrificing the essential compatibility with standard Si technologies. The recent achievements in this application-driven research field, but also the foreseeable problems and limitations, are discussed, and an assessment of the possible role of such heterodevices in future microelectronic circuits is given.