For more than 50 years, research into III–V compound semiconductors has focused almost exclusively on materials grown on (001)-oriented substrates. In part, this is due to the relative ease with which III–Vs can be grown on (001) surfaces. However, in recent years, a number of key technologies have emerged that could be realized, or vastly improved, by the ability to also grow high-quality III–Vs on (111)- or (110)-oriented substrates These applications include: next-generation field-effect transistors, novel quantum dots, entangled photon emitters, spintronics, topological insulators, and transition metal dichalcogenides. The first purpose of this paper is to present a comprehensive review of the literature concerning growth by molecular beam epitaxy (MBE) of III–Vs on (111) and (110) substrates. The second is to describe our recent experimental findings on the growth, morphology, electrical, and optical properties of layers grown on non-(001) InP wafers. Taking InP(111)A, InP(111)B, and InP(110) substrates in turn, the authors systematically discuss growth of both In0.52Al0.48As and In0.53Ga0.47As on these surfaces. For each material system, the authors identify the main challenges for growth, and the key growth parameter–property relationships, trends, and interdependencies. The authors conclude with a section summarizing the MBE conditions needed to optimize the structural, optical and electrical properties of GaAs, InAlAs and InGaAs grown with (111) and (110) orientations. In most cases, the MBE growth parameters the authors recommend will enable the reader to grow high-quality material on these increasingly important non-(001) surfaces, paving the way for exciting technological advances.