We review the unique self-organizing growth mechanisms on planar and patterned high-index semiconductor substrates in atomic hydrogen assisted molecular beam epitaxy (MBE) allowing the reproducible fabrication of dense arrays of quantum wires and quantum dots with large confinement energy and uniform size distribution needed for realistic device applications. On not intentionally patterned substrates, hydrogen induced nanometer-scale self-faceting is directly applied for the formation of conductive quantum wires on GaAs(331)A exhibiting pronounced anisotropy of the electron conductivity. In conventional MBE on intentionally patterned GaAs(311)A substrates the self-limiting formation of a fast growing mesa sidewall is highlighted. The systematic engineering of growth selectivity by appropriate pattern design reveals high flexibility for the fabrication of dense arrays of quantum wires, dot-like nanostructures, and coupled wire-dot arrays with superior control of nanostructure size and position. A new dimension in nanostructure formation is introduced by combining the natural self-faceting on GaAs(311)A substrates with patterned growth in atomic hydrogen assisted MBE producing linear arrays of quantum dots with minimized size fluctuations.