Lattice-mismatched systems are of great scientific and technological interest. Epitaxial growth of Si-based heterostructures allows integrating components made of a variety of semiconductor materials with the well-established Si technology and designing new types of electronic and optoelectronic devices. For some applications, strain in heterostructures has to be relieved, whereas other device structures demand fully strained epitaxial layers. In this review, we concentrate on the ideas providing the basis for different approaches to the control over strain relaxation in lattice-mismatched systems, such as graded-buffer technique, growth of low-temperature buffer layers, selected-area epitaxy, and compliant-substrate concept. The effect of ion irradiation on strain relaxation is also considered. Recent achievements in the reduction of defect density with the use of the reviewed techniques are reported. Potentialities of the various approaches for device applications are briefly discussed. (C) 2004 Elsevier Ltd. All rights reserved.