Stresses and strains in heterostructures have dominated semiconductor research during the last ten years. We review the theory and experimental work on stresses in III - V semiconductor heterostructures in this paper. First large-area lattice mismatched layers (InGaAs and InGaP layers grown on GaAs or InP substrates) and thermally strained layers (GaAs, GaP and InP layers grown directly on Si) are considered. The stresses in large-area epilayer - substrate structures are easy to model because substrate distortion can be neglected and strain in the epilayer is a simple biaxial tetragonal distortion. Calculated splitting of band edges, modification of bandgaps and shifts of Raman modes show good agreement with experiments. Edges of a stripe relax stress in the stripes and induce stress in the substrate. Since stresses in the stripe and the substrate are coupled, calculation of stresses in stripes and substrates is more involved. Recent finite element (FE) calculations of these stresses are discussed in detail and compared with the approximate analytical models and with luminescence and Raman measurements. FE calculations of stresses in buried quantum wires, stressor-induced quantum wires and quantum dots are also discussed. The results of these calculations are used to determine stability and luminescence of the quantum wires and dots and compared with the experimental results. Finally self-organized quantum dots consisting of islands formed during the 3D growth of InAs layers on GaAs are discussed. A possible explanation of the recent observation that they are formed in vertical columns embedded in GaAs is suggested.

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