In this tutorial we review recent progress in the design and growth of epitaxial semiconductor nanostructures in lattice-mismatched material systems. We focus on the Ge on Si model system after pointing out the similarities to III-V and other growth systems qualitatively as well as quantitatively. During material deposition, the first layers of the epitaxial film wet the surface before the formation of strain-driven three-dimensional nanostructures. In particular, we stress that the supersaturation of the wetting layer (WL), whose relevance is often neglected, plays a key role in determining the nucleation and growth of nanodots (NDs), nanodot-molecules and nanowires (NWs). At elevated growth temperatures the Ge reservoir in the planar, supersaturated WL is abruptly consumed and generates NDs with highly homogeneous sizes - a process mainly driven by elastic energy minimization. Furthermore, the careful control of the supersaturated Ge layer allows us to obtain perfectly site-controlled, ordered NDs or ND-molecules on pit-patterned substrates for a broad range of pit-periods. At low growth temperatures subtle interplays between surface energies of dominant crystal facets in the system drive the material transfer from the supersaturated WL into the elongating NWs growing horizontally, dislocation- and catalyst-free on the substrate surface. Due to the similarities in the formation of nanostructures in different epitaxial semiconductor systems we expect that the observation of the novel growth phenomena described in this Tutorial Review for Ge/Si should be relevant for other lattice-mismatched heterostructure systems, too.