Inherent to the nanowire morphology is the exciting possibility of fabricating materials organized at the nanoscale in three dimensions. Composition and structure can be varied along and across the nanowire, as well as within coaxial shells. This opens up a manifold of possibilities in nanoscale materials science and engineering which is only possible with a nanowire as a starting structure. As the variation in composition and structure is accompanied by a change in the band structure, it is possible to confine carriers within the nanowire. Interestingly, this results in the formation of local two, one and zero-dimensional structures from an electronic point of view within the nanowire. This novel palette of nanostructures paves the way toward novel applications in many engineering domains such as lasers, high-mobility transistors, quantum information and energy harvesting. In the present review we summarize and give an overview on recent achievements in the design and growth of advanced quantum structures starting from nanowire templates. The quantum structures presented have been grown by molecular beam epitaxy and correspond to different confinement approaches: quantum wells (2D), quantum wires (1D) and quantum dots (0D).