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We present a method for monolithically integrating mismatched semiconductor materials with Si, coined three-dimensional (3D) heteroepitaxy. The method comprises the replacement of conventional, continuous epilayers by dense arrays of strain- and defect-free, micron-sized crystals. The crystals are formed by a combination of deep-patterning of the Si substrates and self-limited lateral expansion during the epitaxial growth. Consequently, the longstanding issues of crack formation and wafer bowing can be avoided. Moreover, threading dislocations can be eliminated by appropriately choosing pattern sizes, layer thicknesses and surface morphology, the latter being dependent on the growth temperature. We show this approach to be valid for various material combinations, pattern geometries and substrate orientations. We demonstrate that Ge crystals evolve into perfect structures away from the heavily dislocated interface with Si, by using a synchrotron X-ray beam focused to a spot a few hundred nanometers in size and by recording 3D reciprocal space maps along their height. Room temperature photoluminescence (PL) experiments reveal that the interband integrated PL intensity of the Ge crystals is enhanced by almost three orders of magnitude with respect to that of Ge epilayers directly grown on flat Si substrates. Electrical measurements performed on single heterojunction diodes formed between 3D Ge crystals and the Si substrate exhibit rectifying behavior with dark currents of the order of 1 mA/cm2. For GaAs the thermal strain relaxation as a function of pattern size is similar to that found for group IV materials. Significant differences exist, however, in the evolution of crystal morphology with pattern size, which more and more tends to a pyramidal shape defined by stable {111} facets with decreasing width of the Si pillars. © 2013 Elsevier B.V.