Low‐temperature molecular‐beam epitaxy (MBE) in semiconductors is reviewed, with a focus on limited thickness epitaxy (LTE), the regime where crystalline growth over an epitaxial thicknesshepiis followed by a transition to amorphous deposition. The goal is to summarize the main results on this phenomenon, make the connection to other results on low‐temperature MBE, and present the large body of unpublished data onhepi. Since the problem is still not fully understood, all available data that have a bearing on the understanding of the effect are outlined. The scientific questions and practical problems that have driven interest in low‐temperature growth are outlined, and the phenomenon of LTE and the dependence ofhepion the growth conditions are described. The LTE effect is apparently general, but Si(100) is the model system for which most data are available. Breakdown of epitaxy follows a universal curve that is inconsistent with continuous nucleation of the amorphous phase, implying that growth is truly thickness dependent. The epitaxial thickness is thermally activated in substrate temperatureTashepi=h0 exp(−Eact/kBT), withh0following a weak ln(R) orR1/4dependence on deposition rateR.hepiis also strongly influenced by lattice mismatch strain, residual H in the ultrahigh vacuum, and annealing during growth interrupts. Possible mechanisms for LTE are discussed, with particular emphasis on the roles played by H and kinetic roughening, and the key experiments distinguishing these mechanisms are described. Finally, an attempt is made to draw up the best current picture of the phenomenon. It is concluded that roughening provides the fundamental limit to epitaxy at low temperature, but with H contamination playing an important part in controlling surface diffusion: outstanding problems include the rate dependence and the details of the roughening behavior. ©1995 American Institute of Physics.