We review recent advances in the growth of Si1−xGex islands and Ge dots on (001) Si. We first discuss the evolution of the island morphology with Si1−xGex coverage and the effect of growth parameters or post-growth annealing on the shape of the islands and dots. We outline some of the structural and optical properties of Si1−xGex islands and assess progress in the determination of their composition and strain distribution. Finally, we discuss various approaches currently being investigated to engineer Si1−xGex quantum dots and in particular to control their size, density, and spatial distribution. For example, we show how C pre-deposition on Si (001) can influence the nucleation and growth of Ge islands.