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We review recent progress in the growth and characterization of Si 1−x Ge x islands and Ge dots on (001) Si. We discuss the evolution of the island morphology with Si 1−x Ge x coverage, and the effect of growth parameters or post-growth annealing on the shape of islands and dots. We outline some of the structural, vibrational, and optical properties of Si 1−x Ge x islands and review recent advances in the determination of their composition and strain distribution. In particular, we present an analytical electron transmission microscopy study of the Ge spatial distribution in Ge dots and Si/Si 1−x Ge x island superlattices grown by molecular beam epitaxy and ultra-high vacuum chemical vapour deposition. We describe the use of undulated Si 1−x Ge x island superlattices for infrared detection at telecommunication wavelengths. Finally, we discuss various approaches currently being investigated to engineer Si 1−x Ge x quantum dots and, in particular, control their size, density, and spatial distribution. As examples, we show how C pre-deposition on Si(001) can influence nucleation and growth of Ge islands and how low temperature Si homo-epitaxy can lead to a particular surface cusp morphology that may promote dot nucleation.