A major problem challenging specialists in present-day materials sciences is the development of compact, cheap to fabricate heat sinks for electronic devices, primarily for computer processors, semiconductor lasers, high-power microchips, and electronics components. The materials currently used for heat sinks of such devices are aluminum and copper, with thermal conductivities of about 250 W/(m·K) and 400 W/(m·K), respectively. Significantly, the thermal expansion coefficient of metals differs markedly from those of the materials employed in semiconductor electronics (mostly silicon); one should add here the low electrical resistivity metals possess. By contrast, natural single-crystal diamond is known to feature the highest thermal conductivity of all the bulk materials studied thus far, as high as 2,200 W/(m·K). Needless to say, it cannot be applied in heat removal technology because of high cost. Recently, SiC- and AlN-based ceramics have started enjoying wide use as heat sink materials; the thermal conductivity of such composites, however, is inferior to that of metals by nearly a factor two. This prompts a challenging scientific problem to develop diamond-based composites with thermal characteristics superior to those of aluminum and copper, adjustable thermal expansion coefficient, low electrical conductivity and a moderate cost, below that of the natural single-crystal diamond. The present review addresses this problem and appraises the results reached by now in studying the possibility of developing composites in diamond-containing systems with a view of obtaining materials with a high thermal conductivity.